aa r X i v : . [ m a t h . D S ] J a n ASYMPTOTIC TRAJECTORIES OF KAM TORUS
JIANLU ZHANG † , CHONG-QING CHENG ‡ Abstract.
In this paper we construct a certain type of nearly integrable sys-tems of two and a half degrees of freedom: H ( p, q, t ) = h ( p ) + ǫf ( p, q, t ) , ( q, p ) ∈ T ∗ T , t ∈ S = R / Z , with a self-similar and weak-coupled f ( p, q, t ) and h ( p ) strictly convex. For agiven Diophantine rotation vector ~ω , we can find asymptotic orbits towardsthe KAM torus T ω , which persists owing to the classical KAM theory, as longas ǫ ≪ f ∈ C r ( T ∗ T × S , R ) properly smooth.The construction bases on several new approaches developed in [16], wherehe solved the generic existence of diffusion orbits of a priori stable systems. Asan expansion of Arnold Diffusion problem, our result supplies several usefulviewpoints for the construction of preciser diffusion orbits. Introduction
Statement of Main Result.
For a nearly integrable systems(1.1) H ( p, q ) = h ( p ) + ǫf ( q, p ) , ( q, p ) ∈ T ∗ T n , KAM theory assures that the set of KAM tori occupies a rather large-measuredpart in phase space, but it’s still a topologically sparse set. It indicates that for asystem with a freedom not bigger than two degrees, every orbit will be confined inthe ‘cells’ formed by energy surface and KAM tori, and the oscillations of actionvariables do not exceed a quantity of order O ( √ ǫ )[2]. This disproved the ergodichypothesis formulated by Maxwell and Boltzmann: For a typical Hamiltonian on a typical energy surface, all but a setof zero measure of initial conditions, have trajectories covering denselythis energy surface itself.
However, if the number of degrees of freedom n greater than two, the n-dimensionalinvariant tori can not divide each (2n-1)-dimensional energy surface into discon-nected parts and the action variables of trajectories not laying on the tori areunrestrained. So it’s reasonable to modify the ergodic hypothesis and raise: Conjecture 1.1. (Quasi-ergodic Hypothesis[9, 21]) For a typical Hamiltonian ona typical energy surface, there exists at least one dense trajectory.
Date : August 11, 2018.1991
Mathematics Subject Classification.
Primary 37Jxx; Secondary 37Dxx.
Key words and phrases.
Arnold Diffusion, KAM torus, Aubry Mather Theory, VariationalMethod, Asymptotic Trajectory. † Email: [email protected]. ‡ Email: [email protected]. † , CHONG-QING CHENG ‡ The first progress towards this direction is made by V. Arnold [1] in 1964. In hispaper, he constructed a 2.5 degrees of freedom system which has an unperturbednormally hypobolic invariant cylinder (NHIC) and a ‘homoclinic overlap’ structure.This ‘homoclinic overlap’ structure assures the existence of heteroclinic trajectoriestowards different lower-dimensional tori located in the NHIC, and along these het-eroclinic trajectories the slow action variable of shadowing orbits changes of O (1)in a rather long time.We can simplify this mechanism and raise the following: Conjecture 1.2. (Arnold Diffusion[2]) Typical integrable Hamiltonian systemswith n degrees of freedom ( n ≥ .
5) is topologically instable: through an arbitrar-ily small neighborhood of any point there passes a phase trajectory whose slowvariables drift away from the initial value by a quantity of order O (1).Celebrated progress has been made through the past twenty years and we cangive a quite positive answer to this Conjecture 1.2. For results of a priori unstablecase, the readers can see [14, 15, 19, 49] and [16, 39, 31] of a priori stable case.But a definite answer whether Conjecture 1.1 is right or wrong is still far from thereach of modern dynamical theory.One instinctive idea towards Quasi-ergodic Hypothesis is to use the same methodin solving Conjecture 1.2 to construct trajectories to fill the topologically open-densecomplement of KAM tori. So to find asymptotic trajectories of KAM tori is thefirst difficulty we must overcome. The first exploration was made by R. Duady[20]: Theorem 1.3.
For a fixed Diophantine rotation vector ~ω , there exists a nearlyintegrable system H ǫ ( p, q ) (cid:12)(cid:12) ( p,q ) ∈ T ∗ T which is C ∞ -approached to an integrable sys-tem H ( p ) , such that for any open neighborhood U n (cid:12)(cid:12) n ∈ N of T ω , there exists onetrajectory γ n of system H ǫ ( p, q ) entering U n from the place O (1) far from T ω . From this theorem, we could deduce that KAM torus is of Lyapunov instability.But as n → ∞ , γ n is different from each other. So his construction is invalid tofind asymptotic orbits of KAM torus.We can generalize Arnold’s construction of [1] to a certain type of nearly inte-grable systems, which is known by a priori unstable ones. This condition actuallyassures that the existence of NHIC (Normally Hyperbolic Invariant Cylinder) withconsiderable length. Based on the celebrated theory developed by J. Mather, [14]and [15] first found the generic existence of diffusion orbits in this case from thevariational view. More generalized nearly integrable systems are called a priori stable systems. We will face new difficulties in solving this case comparing to apriori unstable one:(1) non-existence of long-length NHIC. Complicated resonance-relationship di-vides the 1-resonance lines into short segments, so we can’t just use theoverlap mechanism to find O (1) diffusion orbits.(2) Coming out of 2-resonance. NHICs corresponding to different 1-resonancesegments are separated by the chaotic layer caused by 2-resonance. Weneed to find trajectories in this layer to connect them (see figure 1.1). SYMPTOTIC TRAJECTORIES OF KAM TORUS 3
Figure 1.
Since Diophatine vector ω is non-resonant, we have to overcome these two diffi-culties in finding asympytotic orbits. The first announcement of a priori stablecase was given by J. Mather in 2003. He defined a conception ‘cusp residue’ tomeasure the ‘size’ of a set in topological space. Later, C-Q. Cheng verified the cuspgenericity of diffusion orbits in a priori stable case. In [16], he proposed a plan toovercome the difficulties caused by 2-resonance: Around the lowest flat F . = { c ∈ H ( T , R ) (cid:12)(cid:12) α H ( c ) = min α H } there existsan incomplete intersection annulus A ⊆ H ( T , R ) whose width we couldprecisely calculate and the Ma˜n´e set ˜ N ( c ) ranges as a broken lamina-tion structure, ∀ c ∈ A . Besides, the cohomology classes corresponding toNHICs could plug into A . Based on this idea, we can connect different NHICs in this annulus A withMather’s mechanism diffusion orbits discovered in [36]. These orbits can be con-nected with the ’Arnold’ mechanism diffusion orbits of NHICs and we succeed toconstruct O (1) diffusion orbits in a priori stable case.From now on we only consider the case of 2 . − degrees of freedom. As a specialcase of a priori stable systems, finding asymptotic trajectories of KAM torus willface another new difficulty: infinitely many changes of 1-resonance lines will beinvolved in. Here we can give a rough explanation on this. From [16, 39] weknow that the ‘cusp genericity’ is caused by the restrictions of hyperbolic strengthon different 1-resonant lines. Since ω is non-resonant, it’s unavoidable to faceinfinitely many 1-resonant lines. These lines cause infinite times ‘cusp remove’ tothe perturbed function space C r ( T ∗ T × S , R ) on the contrary. So we only havea ‘porous’ set P r . = { f ∈ C r ( T ∗ T × S , R ) (cid:12)(cid:12) k f k C r ≤ } left, of which we havethe chance to find asymptotic trajectories. Recall that this set P r is not open in C r ( T ∗ T × S , R )! Theorem 1.4.
For nearly integrable systems written by (1.2) H ( p, q, t ) = h ( p ) + ǫf ( p, q, t ) , ( p, q, t ) ∈ T ∗ T × S , ǫ ≪ , JIANLU ZHANG † , CHONG-QING CHENG ‡ here h ( p ) is strictly convex, ∇ h (0) = ω , and D h (0) is strictly positively definite.For a fixed Diophantine vector ˜ ω = ( ω, ∈ R , we could find ǫ = ǫ ( ω, D h ( o )) such that for ǫ ≤ ǫ and f ( q, p, t ) ∈ P r ( r ≥ with self-similar and weak-coupledstructures, of which we can find asymptotic trajectories of KAM torus T ω .Remark . Here the self-similar and weak-coupled structures are proposed tothe Fourier coefficients of f ( q, p, t ). To avoid the collapse of infinitely many timescusp-remove, we should control the speed of decline of hyperbolicity along theresonance lines tend to ω , and then the Fourier coefficients are involved in. Laterwe will see that the Fourier coefficients corresponding to different resonant lines areindependent from each other. We could benefit from this and raise a self-similarstructure to simplify our treatment of infinite resonance relationships to finite ones.For a system of a form (1.2), we can ensure the persistence of T ω as long as ǫ issufficiently small. Then the following holds: Theorem 1.6. [13]
There exists a smooth exact symplectic transformation T ∞ f : D → D , where D ⊂ T ∗ T × S is a small neighborhood of { } × T × S . For ǫ ≤ ǫ and under this transformation, we can convert Hamiltonian (1.2) to (1.3) H ( p, q, t ) = h ( p ) + h p t , f ( q, t ) p i + O ( p ) , ( q, p, t ) ∈ D , here h p t , f ( q, t ) p i is a quadratic polynomial with p t = ( p , p ) , ∇ h (0) = ω , and D h (0) is strictly positively definite.Proof. It’s a direct cite of Lemma (6.1) in [13] for details. (cid:3)
Moreover, we can use finite steps of ‘Birkhoff Normal Form’ transformations toraise the order of polynomial and get
Lemma 1.7.
There exists another smooth exact symplectic transformation R ∞ f : D → D under which system (1.3) can be changed into (1.4) H ( q, p, t ) = h ( p ) + p σ f ( q, t ) + O ( p σ +1 ) , ( q, p, t ) ∈ D , with a sufficiently large σ ∈ Z + and f ( q, t ) ∈ C r ( T × S , R )( r ≥ × .Remark . In the above theorem we omit the small number ǫ which assures theexistence of KAM torus T ω , since we can restrict diam D much smaller than ǫ . Wejust need to find asymptotic trajectories in this domain. From now on, we willwrite h p t , f ( q, t ) p i , p i , · · · , p i | {z } σ − as p σ f ( q, t ) for short without confusion.Based on Theorem 1.6 and Lemma 1.7, we could convert Theorem 1.4 into thefollowing Theorem 1.9. (Main Result)
For the system of a form (1.4), we can find proper f ( q, t ) ∈ C r ( T × S , R ) with a self-similar and weak-coupled structure of which thereexists at least one asymptotic trajectory to the KAM torus T ω . At last, we sketch out our plan: to find the proper f ( q, t ), we need a list of ‘rigid’conditions to be satisfied. Owing to these conditions, we can give a ‘skeleton’ of f ( q, t ) which is not easy to be destroyed. then we use ‘soft’ generic perturbationsto construct diffusion orbits which finally tends to T ω . Here, ‘soft’ means the per-turbations can be chosen arbitrarily small and arbitrarily smooth, which is known SYMPTOTIC TRAJECTORIES OF KAM TORUS 5 from [14], [15] and [16].From the proof the readers can see that the frame we made in order to get theasymptotic orbits is ‘firm’ enough and small perturbations can’t destroy it. Ourmethod is neither the same with the way V. Kaloshin and M. Saprykina used in[29], nor the same with the way P. Calvez and R. Douady used in [10](in theirpapers they considered some close problems with ours). Our new approach benefitsus with the chance to find more systems satisfying our demand.We also recall that other two papers related with our result: one is [30] in 2010and the other is [28] in 2004. The latter one considered the asymptotic trajectoriesof resonant elliptic points, which is different from our situation and only finitelymany resonant lines are involved in.This paper is our first step to find preciser diffusion orbits in general systems.There’s still a long way to go for the target of giving a rigorous answer to thequasi-ergodic hypothesis, and it’s still open to find alternative mechanisms to con-struct diffusion orbits. Interestingly, T. Tao found an example of cubic defoucusingnonlinear Schr¨odinger equation of which energy transports to higher frequencies in[48]. His construction shares some similarities with Arnold Diffusion. Moreover, aself-similar resonant structure with special arithmetic properties is also applied inhis construction. Aware of these, we are confident that it must be a hopeful direc-tion to apply our diffusion mechanisms to PDE problems. Recently, M. Guardiaand V. Kaloshin have made some progress in this domain[25].1.2.
Outline of the Proof.
We just need to prove Theorem 1.9 which is a specialform of Theorem 1.4 via a C ∞ symplectic transformation. As we can see, system(1.4) is a Tonelli Hamiltonian, of which T ω = { p = 0 } × T × S is actually un-perturbed. We could find a skeleton of infinitely many resonant lines { Γ ωi } ∞ i =1 inthe frequency space, along which Γ ωi approaches to Diophantine vector ω as i → ∞ (See figure 2). We could divide this skeleton into accord-ing to different resonant relationships. Each of them we need different mechanismsto deal with.For the former two cases, to supply the NHICs with enough hyperbolicity, weneed several ‘rigid’ conditions U2,3 for the Fourier coefficients of f ( q, t ) corre-sponding to the current resonant line Γ ωi of considerations. For the 2-resonant caseΓ ωi ∩ Γ ωi +1 , a ‘weak-coupled’ structure can be available by properly choosing themixed Fourier coefficients according to Γ ωi and Γ ωi +1 . This simplifies the dynamicbehaviors of 2-resonance greatly. Also several ‘rigid’ conditions U → ωi , which we call sub 2-resonant points. Since there isn’t any transition between different resonant linesat these points, our diffusion orbits just need to cross them and go on along the JIANLU ZHANG † , CHONG-QING CHENG ‡ Figure 2.
NHICs according to 1-resonant lines. To achieve this, rigid condition U3 is needed.Recalls that infinitely many resonant relationships are considered in our case,so these ‘rigid’ conditions have to be uniformly satisfied according to all of theseresonant lines. That’s why we mark the ‘rigid’ conditions with a letter ‘U’. It’s thecost to avoid the collapse caused by infinite times cusp remove. But the self-similarstructure simplifies the complexity and gives us a universal treatment for all theresonant relationships.Now we explain why these ‘rigid’ conditions can be satisfied without conflict andgive a sketch of the construction. We begin with such a Tonelli Hamiltonian:(1.5) H ( q, p, t ) = h ( p ) + p σ f ( q, t ) , ( q, p, t ) ∈ D . • First, we choose a proper resonant plan { Γ ωi } ∞ i =1 ,i ∈ N which approximate ω steadily. • Second, along these resonant lines, we can transform system (1.5) to a Reso-nant Normal Form H = h + Z + R with finite KAM iterations in a neighborhood of {B (Γ pi , δ i ) × T × S } ∞ i =1 . Here Z = [ f ] ω ∗ is the average term corresponding to the cur-rent frequency ω ∗ . All these conditions and structural demands can all be satisfiedby Z as long as the Z sequence of Fourier coefficients { f ( k ,k ,k ) ∈ R } k i ∈ Z ,i =1 , , are properly chosen, where ~k = ( k , k , k ) ∈ Λ ω ∗ subspace of Z . We can see thatdifferent resonant lines will decide different subspaces of Z which are independentfrom each other. This point is very important for our case. • Third, at the 2-resonance Γ ωi ∩ Γ ωi +1 , we need to connect different NHICs oftheir bottoms. A weak-coupled structure can be available by the mixed terms ofFourier coefficients according to Γ ωi ∩ Γ ωi +1 . This structure doesn’t damage thehyperbolicity of NHICs and are strong enough to supply us a chaotic layer withsufficient width, which we called incomplete intersection annulus here. In thispart we mainly used the same method proposed in [16]. SYMPTOTIC TRAJECTORIES OF KAM TORUS 7
It’s remarkable that this ‘weak-coupled’ structure reduces the complexity of dy-namical behaviors greatly at the 2-resonant domain, which can be considered as anapplication of Melnikov’s approach. We take [50] for a convenient reference. • At last, since all the ‘rigid’ conditions have been satisfied, we can perturbsystem (1.5) step by step to find diffusion trajectories which extend towards to T ω gradually. Based on the matured methods of genericity and regularity developedby [16, 14, 15] and [32], the perturbation functions are very ‘soft’, i.e. they can bemade arbitrarily small and smooth. Here we use a list of perturbations { f j } ∞ j =1 ,j ∈ N to modify system (1.5) and get a system of a form (1.4), for which we indeed getan asymptotic trajectory of T ω .The paper is organized as follows. In the rest part of this section we will give asketch of the Mather Theory and Fathi’s weak-KAM method, also several proper-ties about global elementary weak-KAM solutions in the finitely covering space[16].In Section 2, we give a resonant plan { Γ ωi } ∞ i =1 ,i ∈ N to approximate ω and get itsfine properties. Section 3 supplies a Stable Normal Form with finite KAM itera-tions which is unified for all the Γ i . In this part all the ‘rigid’ conditions couldbe raised naturally. In Section 4, we prove the existence of NHICs in every caseseparately, 1-resonance, transitional segments from 1-resonance to 2-resonance andthen 2-resonance. In the later two cases, a homogenized method is involved. Inthe 2-resonance case, a weak-coupled structure is used and we could give a preciseestimate about the lowest positions where the bottoms of NHICs could persist. Be-sides, we get the conclusion that Aubry sets just locate on these NHICs. In Section5, the existence of incomplete intersection annulus of c-equivalence is establishedand its width can also be precisely estimated. In Section 6, we recall the approachto get two types of locally connecting orbits by modifying the Lagrangian. Thispart is mainly based on genericity and regularity of [14, 15, 32] and [16]. Withthese preliminary works, we get our asymptotic trajectories by a list of ‘soft’ per-turbations in Section 7. Therefore, we finish our construction of system (1.4) andget our main conclusion.1.3. Brief introduction to Mather Theory and properties of weak KAMsolutions.
In this subsection we will give a profile of the tools we used in this paper: MatherTheory and weak KAM theorem. Recall that the earliest version of weak KAMtheory which Fathi gave us in [22] mainly concerns the autonomous Lagrangians,but most of the conclusions are available for the time periodic case.
Definition 1.10.
Let M be a smooth closed manifold. We call L ( x, ˙ x, t ) ∈ C r ( T M × S , R ) ( r ≥
2) a
Tonelli Lagrangian if it satisfies the following condi-tions: • Positive Definiteness:
For each ( x, ˙ x, t ) ∈ T M × S , the Lagrangian func-tion is strictly convex in velocity, i.e. the Hessian matrix ∂ ˙ x ˙ x L is positivelydefinite. • Superlinearity:
L is fiberwise superlinear, i.e. for each ( x, t ) ∈ M × S ,we have L/ k ˙ x k → ∞ as k ˙ x k → ∞ . JIANLU ZHANG † , CHONG-QING CHENG ‡ • Completeness:
All the solutions of the Euler-Lagrangian (E-L) equationcorresponding to L are well-defined for t ∈ R .For such a Tonelli Lagrangian L , the variational minimal problem of γ ∈ C ac ([ a, b ] , M )with fixed end points γ ( a ) = x , γ ( b ) = y is well posed as: A L ( γ ) = inf γ ∈ C ac ([ a,b ] ,M ) γ ( a )= x,γ ( b )= y Z ba L ( γ ( t ) , ˙ γ ( t ) , t ) dt. Then we can see that if γ is the critical curve of this variational problem, then itmust satisfy the Euler-Lagrangian equation:(1.6) ddt ∂ ˙ x L ( γ ( t ) , ˙ γ ( t ) , t ) = ∂ x L ( γ ( t ) , ˙ γ ( t ) , t ) , t ∈ [ a, b ] . We call a curve γ ⊂ M is a solution of E-L equation if ∀ a, b ∈ R with a < b , γ ( t ) satisfies (1.6) for t ∈ [ a, b ]. Once γ is a solution of L , we actually see that γ ∈ C r ( R , M ) from the Tonelli Theorem [40]. We can define a flow map of φ tL : T M × S → T M × S by φ tL ( x, v, s ) = ( γ x,v ( t ) , ˙ γ x,v ( t ) , t + s mod 1) ∈ T M × S , ∀ t ∈ R , s ∈ S , where ( γ x,v (0) , ˙ γ x,v (0) , s ) = ( x, v, s ) and γ x,v is a solution of L . Then we can gen-erate a φ L − invariant probability measure µ γ x,v by γ x,v with the following ergodicTheorem(1.7) lim T →∞ T Z T L ( γ x,v ( t ) , ˙ γ x,v ( t ) , t ) dt = Z T M × S L ( x, v ) dµ γ x,v . We denote the set of all the φ L − invariant probability measures by M inv ( L ). Remark . Here we need to make a convention once for all. A continuous map γ : I → M is called a curve , where I ⊂ R is an interval either bounded orunbounded, open or closed. For the time-dependent case, ∪ t ∈ I ( γ ( t ) , t ) is consideredas a curve as well. We call ( γ, ˙ γ, t ) : I → T M × R an orbit , or a trajectory , iffit’s invariant under the flow map φ tL .Let η c ( x ) dx be a closed 1-form on M , with [ η c ] = c ∈ H ( M, R ). Then L c . = L ( x, v, t ) − h η c , v i , ( x, v, t ) ∈ T M × S is also a Tonelli Lagrangian and we can see that any solution of L is also a solutionof L c , vise versa. This supplies us with a chance to distinguish measures of M inv ( L )by cohomology class.We define the α − function of L by(1.8) α L ( c ) = − inf µ ∈ M inv ( L ) Z T M × S L − η c dµ. It’s a continuous, convex and super-linear function. We call the minimizer of abovedefinition a c-minimizing measure , and the set of all c-minimizing measures canbe written by M inv ( c ). It’s a convex set of the space of all the probability mea-sures, under the weak* topology. If µ c is a extremal point of M inv ( c ), it must bean ergodic measure with its support a minimal set for φ tL . SYMPTOTIC TRAJECTORIES OF KAM TORUS 9
For µ ∈ M inv ( L ), there exists a ρ ( µ ) ∈ H ( M, R ) according to it and h ρ ( µ ) , [ η c ] i = Z T M × S η c dµ, for every closed 1-form η c with [ η c ] = c ∈ H ( M, R ). Here h , i denotes the canonicalpairing between homology and cohomology. Then we have the following conjugatedfunction(1.9) β L ( h ) = inf µ ∈ M inv ( L ) ρ ( µ )= h Z T M × S Ldµ, h ∈ H ( M, R ) . It’s also continuous, convex, and super-linear [37]. Similarly, we can define theset of all the minimizers of above formula by M inv ( h ). Let D − α L ( c ) be the sub-differential set of α L ( c ) at c and D − β L ( h ) be the sub-differential set of β L ( h ) at h .Then we have the following properties • β L ( h ) + α L ( c ) = h c, h i , ∀ c ∈ D − β L ( h ), h ∈ D − α L ( c ). • ∀ µ h ∈ M inv ( h ), we have µ h ∈ M inv ( c ) with c ∈ D − β L ( h ). • ∀ µ c ∈ M inv ( c ), we have µ c ∈ M inv ( ρ ( µ c )).The union set of all the c-minimizing measures’ support is the so-called Matherset , which is denoted by f M L ( c ). Its projection to M × S is the projected Matherset M L ( c ). From [37] we know that π − (cid:12)(cid:12) M ( c ) : M × S → T M × S is a Lipschitzgraph, where π is the standard projection from T M × S to M × S .Sometimes, the Mather set is too ‘small’ to handle with, so larger invariant setsshould be involved in. We define(1.10) A c ( γ ) (cid:12)(cid:12) [ t,t ′ ] = Z t ′ t L ( γ ( t ) , ˙ γ ( t ) , t ) − h η c ( γ ( t )) , ˙ γ ( t ) i dt + α ( c )( t ′ − t ) , (1.11) h c (( x, t ) , ( y, t ′ )) = inf ξ ∈ C ac ([ t,t ′ ] ,M ) ξ ( t )= xξ ( t ′ )= y A c ( ξ ) (cid:12)(cid:12) [ t,t ′ ] , where t, t ′ ∈ R with t < t ′ , and(1.12) F c (( x, τ ) , ( y, τ ′ )) = inf τ = t mod 1 τ ′ = t ′ mod 1 h c (( x, t ) , ( y, t ′ )) , where τ, τ ′ ∈ S . Then a curve γ : R → M is called c-semi static if F c (( x, τ ) , ( y, τ ′ )) = A c ( γ ) (cid:12)(cid:12) [ t,t ′ ] , for all t, t ′ ∈ R and τ = t mod 1, τ ′ = t ′ mod 1. A semi static curve γ is called c-static if A c ( γ ) (cid:12)(cid:12) [ t,t ′ ] + F c (( γ ( t ′ ) , t ′ ) , ( γ ( t ) , t )) = 0 , ∀ t, t ′ ∈ R . The
Ma˜n´e set which is denoted by e N ( c ) ⊂ T M × S is the set of all the c-semistatic orbits. Theorem 1.12. (Upper semicontinuity [14, 15] ) The set-valued function ( c, L ) → e N ( c ) is upper semicontinuous. † , CHONG-QING CHENG ‡ We can similarly define the
Aubry set by the set of all the c-static orbits, whichcan be written by ˜ A ( c ). Then we have f M ( c ) ⊂ ˜ A ( c ) ⊂ e N ( c ) . Note that from now on we can omit the subscripts ‘inv’, ‘L’ for short. We alsodenote the projected Ma˜n´e set by N ( c ) and the projected Aubry set by A ( c ). From [38] we can see that π − : A ( c ) ⊂ M × S → ˜ A ( c ) ⊂ T M × S is also aLipschitz graph. Let(1.13) h ∞ c (( x, s ) , ( x ′ , s ′ )) = lim inf s = t mod 1 s ′ = t ′ mod 1 t ′ − t →∞ h c (( x, t ) , ( x ′ , t ′ )) , then we have h ∞ c (( x, τ ) , ( x, τ )) = 0 , ∀ ( x, τ ) ∈ A ( c ) . We can further define a pseudo metric on A ( c ) by d c (( x, τ ) , ( x ′ , τ ′ )) = h ∞ c (( x, τ ) , ( x ′ , τ ′ )) + h ∞ c (( x ′ , τ ′ ) , ( x, τ ))and then get an equivalent relationship: ( x, t ) ∼ ( x ′ , t ′ ) implies d c (( x, t ) , ( x ′ , t ′ )) =0. Let A ( c ) / ∼ be the quotient Aubry set, and the element of A ( c ) / ∼ is called an Aubry class , which can be written by A i ( c ). We can see that A ( c ) = ∪ i ∈ Λ A i ( c ),where Λ is an index set. We can define the Barrier function between differentAubry classes by B c,i,j ( z, r ) = h ∞ c (( x, t ) , ( z, r )) + h ∞ c (( z, r ) , ( y, s )) − h ∞ c (( x, t ) , ( y, s )) , ∀ ( x, t ) ∈ A i ( c ) , ( y, s ) ∈ A j ( c ) , ( z, r ) ∈ M × S . (1.14) Remark . In some case, we need to consider the properties of curves in thefinite covering space ¯ M . Analogously, we can copy the conceptions of c-semi staticcurve and c-static curve on it, and take N ( c, ¯ M ) and A ( c, ¯ M ) as the according sets.We can see that N ( c ) ( N ( c, ¯ M ) and π A ( c, ¯ M ) = A ( c ) with π : ¯ M → M theprojection map. From [17] we can see that different Aubry class of A ( c, ¯ M ) canalways be connected by c-semi static curves of N ( c, ¯ M ). This point plays a veryimportant role in our diffusion mechanism.In this following part, we’ll give a survey about Fathi’s weak KAM theory, whichcan be seen as a Hamiltonian version of Mather theory. Definition 1.14.
We call a time-periodic system H ( x, p, t ) : T ∗ M × S → R Tonelli Hamiltonian , if it satisfies the following: • Positive Definiteness:
For each ( x, p, t ) ∈ T M × S , the Hamiltonianis strictly convex in momentum, i.e. the Hessian matrix ∂ pp H is positivedefinite. • Superlinearity: H is fiberwise superlinear, i.e. for each ( x, t ) ∈ M × S ,we have H/ k p k → ∞ as k p k → ∞ . • Completeness:
All the solutions of the Hamiltonian equation correspond-ing to H are well-defined for t ∈ R .We can associate to the Hamiltonian H a Tonelli Lagrangian L : T M × S → R by the Legendre transformation:(1.15) L ( x, v, t ) = sup p ∈ T ∗ x M h p, v i − H ( x, p, t ) . SYMPTOTIC TRAJECTORIES OF KAM TORUS 11
Then ∂ p H : T ∗ M × S → T M × S become a diffeomorphism, whose inverse mapis given by ∂ v L : T M × S → T ∗ M × S . Note that the right side of (1.15) getits maximum for v = ∂ p H ( x, p ). We can see that once ( γ, ˙ γ ) : R → T M satis-fies the E-L equation, then ( γ ( t ) , ∂ v L ( γ ( t ) , ˙ γ ( t ) , t )) : R → T ∗ M must satisfies theHamiltonian equation of H , i.e. ( γ ( t ) , p ( t )) is a trajectory of the flow map φ tH with p ( t ) = ∂ v L ( γ ( t ) , ˙ γ ( t ) , t ).Let η c be a closed 1-form of M with [ η c ] = c , then we can make L c . = L − η c and H c ( x, p, t ) . = H ( x, p + η c ( x ) , t ) , ∀ ( x, p ) ∈ T ∗ M, with H ( x, p, t ) the corresponding Hamiltonian of L ( x, v, t ). We can define such aLax-Oleinik mapping on C ( M × S , R ) by(1.16) T − c,t u ( x, s ) = min y ∈ M [ u ( y, s − t ) + h c (( y, s − t ) , ( x, s ))] , where u ( x, s ) is a fixed continuous function of M × S . Then we can get the followingfixed point of Lax-Oleinik mapping by u − c ( x, s ) . = lim inf t →∞ T − c,t u ( x, s ) . We call this u − c a weak KAM solution of H system[6]. For a fixed t ∈ [0 , u − c ( · , t ) is semi concave with linear modulus of x ∈ M (SCL( M )).This is because the uniformly convexity of H . Definition 1.15. [11] We say a function u : M → R is semi concave with linearmodulus if it’s continuous and there exists C ≥ u ( x + h ) + u ( x − h ) − u ( x ) ≤ C k h k , for all x, h ∈ M . The constant C is called the semi concavity constant of u .As a SCL(M) function is C − differentiable almost everywhere, then we have thefollowing(1.17) ∂ t u − c ( x, t ) + H ( x, du − c ( x, t ) + c, t ) = α L ( c ) , a.e. ( x, t ) ∈ M × S . Actually, u − c ( x, t ) is a viscosity solution of above Hamiltonian-Jacobi equation,which can be seen from [22].For a c-semi static orbit ( γ ( t ) , ˙ γ ( t )), we can see that u − c is differentiable at γ ( t )and c + du − c ( γ ( t )) = ∂ v L ( γ ( t ) , ˙ γ ( t ) , t ). Besides, ( γ ( t ) , c + du − c ( γ ( t ))) is a Hamiltonianflow of φ tH ( γ (0) , du − c ( γ (0))), and ∂ t u − c ( γ ( t )) + H ( γ ( t ) , c + du − c ( γ ( t ) , t ) , t ) = α L ( c ) , ∀ t ∈ R . So we can define e N H ( c ) = [ γ ( t ) ∈N L ( c ) t ∈ R ( γ ( t ) , c + du − c ( γ ( t ) , t ) , t )and e A H ( c ) = [ γ ( t ) ∈A L ( c ) t ∈ R ( γ ( t ) , c + du − c ( γ ( t ) , t ) , t )by the conjugated Ma˜n´e set and Aubry set. Sometimes, we can change the sub-script of α L ( c ) to H , as long as H is conjugated to L . † , CHONG-QING CHENG ‡ On the other side, Let ˜ L ( x, v ) . = L ( x, − v ) be the symmetrical Lagrangian of L , then we have a similar Lax-Oleinik mapping ˜ T − c,t on C ( M × S , R ) and ∀ u ∈ C ( M × S , R ),(1.18) ˜ u − c ( x, s ) . = lim inf t →∞ ˜ T − c,t u ( x, s )exists for all ( x, s ) ∈ M × S . It’s the weak KAM solution of ˜ H , which is of theform ˜ H ( x, p, t ) = sup v ∈ T x M h p, v i − ˜ L ( x, v, t ) . Take a special function w = − u ( x, t ) ∈ C ( M × S , R ) into the semi-group and getthe inferior limit as (1.18), then we can see that u + c . = − ˜ w − c satisfies(1.19) ∂ t u + c ( x, t ) + H ( x, du + c ( x, t ) + c, t ) = α L ( c ) , a.e. ( x, t ) ∈ M × S . Definition 1.16. ( γ ( t ) , ˙ γ ( t )) : R → T M is called backward c-semi static orbit ,if there exists a T − ∈ R such that F c (( γ ( t ) , τ ) , γ ( t ′ ) , τ ′ )) = A c ( γ ) (cid:12)(cid:12) [ t,t ′ ] , holds for all t < t ′ ∈ ( −∞ , T − ]. Analogously, ( γ ( t ) , ˙ γ ( t )) : R → T M is called forward c-semi static orbit , if there exists a T + ∈ R such that F c (( γ ( t ) , τ ) , γ ( t ′ ) , τ ′ )) = A c ( γ ) (cid:12)(cid:12) [ t,t ′ ] , holds for all t < t ′ ∈ [ T + , ∞ ). Here τ = t mod 1 and τ ′ = t ′ mod 1. Theorem 1.17. [22]
Let ( x, t ) ∈ M × S be a differentiable point of u − c (or u + c ). As the initial condition, ( x, du − c ( x )) ( ( x, du + c ( x )) ) will decide a unique tra-jectory of H by ( x − ( t ) , p − ( t )) : R → T ∗ M , ( ( x + ( t ) , p + ( t )) : R → T ∗ M ) with ( x − (0) , p − (0)) = ( x, du − c ( x )) ( ( x + (0) , p + (0)) = ( x, du + c ( x )) ). The correspondingorbit ( x − ( t ) , ∂ p H ( x − ( t ) , p − ( t ) , t )) : R → T M ( ( x + ( t ) , ∂ p H ( x + ( t ) , p + ( t ) , t )) : R → T M ) is backward c-semi static on ( −∞ , (forward c-semi static on [0 , + ∞ ) ). From [16] we know, in a proper covering space ¯ M , A H ( c, ¯ M ) may have severalclasses, even though A H ( c ) is of uniquely class. These different classes of A H ( c, ¯ M )are disjoint from each other[38], which can be written by A iL ( c, ¯ M ), i ∈ Λ. Thenwe can find a sequence of Tonelli Hamiltonians { H ij : T ∗ ¯ M × S → R } ∞ j =1 toapproximate H under the C r − norm, such that A iH ( c, ¯ M ) is the unique Aubryclass of H ij . Accordingly, we can find a sequence of weak KAM solutions of { u − c,i,j : ¯ M × S → R } ∞ j =1 which converges to a special weak KAM solution u − c,i ofsystem H in ¯ M . That’s our elementary weak KAM solution of class A i ( c, ¯ M ).Analogously, we get all the elementary weak KAM solutions { u − c,i } i ∈ Λ .With the help of this definition, we can translate our Barrier function B c,i,j in¯ M × S into a simpler form:(1.20) B c,i,j ( z, r ) = u − c,i ( z, r ) − u + c,j ( z, r ) , ∀ ( z, r ) ∈ ¯ M × S . SYMPTOTIC TRAJECTORIES OF KAM TORUS 13 Choose of resonant plan and Fourier properties of functions
For convenience, we first make a convention on the symbol system once for all.Recall that the system (1.5) is of the form H ( p, q, t ) = h ( p ) + p σ f ( q, t ) , ( p, q, t ) ∈ D ⊆ T ∗ T × S , where p σ f ( q, t ) is actually a polynomial of multi-variables p = ( p , p ). Here h (0) =0, ∇ h (0) = ~ω , D h (0) is strictly positively definite and k D h (0) k ∼ O (1). f ( q, t ) ∈ B (0 , c ) ⊆ C r ( T × S , R ), where c ∼ O (1) is a fixed constant. ~ω is a Diopantinefrequency of index ( τ, C ), i.e. ∀ ~k ∈ Z , |||h ~k, ~ω i||| ≥ C | ~k | n − τ , where | ~k | = max {| k | , | k |} , ~k = ( k , k ) and ||| x ||| = x − [ x ] is the reminder partof x . We add a subscript ‘0’ to the Diophantine frequency ω to avoid confusion inthe following. Notice that T = R / π Z and S = R / Z in our situation.We denote the norm k · k C r of C r ( T × S , R ) by k f ( q, t ) k . = P r | ~α | =0 k f ( q, t ) k C ,where | ~α | . = | α | + | α | + | α | and k · k C is the uniform norm. Lemma 2.1. ∀ f ( q ) ∈ C r ( T , R ) , q = ( q , q , q ) we have: (1) k f ~k k C ≤ (2 π | ~k | ) − r · k f k C r , here ~k = ( k , k , k ) ∈ Z and | · | is denotedas above. (2) κ . = P ~k ∈ Z | ~k | − − is a constant of O (1) , here ‘3’ can be replaced by adimensional argument n . (3) R K f . = P | ~k ≥ K | f ~k exp πi h ~k,q i , then k R K f k C ≤ κ K − r +3+3 k f k C r .Proof. (1) Since f ~k = π R π f ( q ) exp − πi h ~k, q i dq , we can easily get the esti-mate by r-times integral by parts.(2) The number of ~k satisfying | ~k | = l is less than 2 l − , so P k ∈ Z | k | − − ≤ P l ∈ N l − l ≤ P l ∈ N l < ∞ . Same result for T n can be get withreplacing 3 by n .(3) We also use a skill of integral by parts: k R K f k C ≤ P | k |≥ K | k | − r +2 k f k C r ≤ K − r +3+3 k f k c r P | k |≥ K | k | − − , then use the result of (2). By the way, r ≥ (cid:3) Notice:
For simplicity, we often omit the vector symbol ~ · . Later on , we alsoneed a norm k · k C r , B ( p ∗ ,δ ) on some subdomain of phase space T ∗ T × S , whichcould be defined in the same way as above, except the action variables p = ( p , p )added. Sometimes we denote the norm by k · k C r , B or k · k C r ,δ for short, as long asthere’s no ambiguity.From the aforementioned Lemma, ∀ f ∈ C r ([0 , , R ), there will be a unique Z -real number sequence { f k } k ∈ Z corresponding to it and the rate of decay of | f k | hasbeen given by (1) as | k | → ∞ . Conversely, a Z -real sequence { f k } k ∈ Z satisfyingLemma 2.1 will determine a function f in C r ([0 , , R ). Definition 2.2.
We denote the space of Z -real sequences by C and the subspaceof C which could decide C r functions by C r . † , CHONG-QING CHENG ‡ (1) A linear subspace of Z is called a Lattice , which is written by Λ. If wecan find a group of irreducible base vectors generating Λ, then Λ is calleda greatest Lattice and is denoted by Λ max . We denote the space of allLattices by L .(2) We call the linear operator F a Pickup of C r , if F : C r × L → C k via ( f k , Λ) → ¯ f k , where k ∈ Z and ¯ f k = (cid:26) f k , k ∈ Λ0 , k / ∈ Λ(3) We call the linear operator G a Shear , if G : C r × L × R → C k via ( f k , Λ , K ) → ¯ f k,K , where k ∈ Z and ¯ f k,K = (cid:26) f k , k ∈ Λ | k | ≥ K, , else Now we make use of these definitions to get our resonant plan. Without loss ofgenerality, We can assume ω = ( ω , , ω , ) ∈ [0 , × [0 , l ∈ N \ { } , we canget a l -partition of [0 , , i.e. l little squares which are diffeomorphic to [0 , l ] .We continue this process m -times and get squares of length l m . We can always picka proper l m -lattice point ω m with dist( ω m , ω ) ∈ [ √ l m +1 , √ l m ] for each step, m ∈ N .Additionally, we have dist ( ω m , ω ) < dist ( ω n , ω ) , ∀ n < m. The following demonstration will give the readers a straightforward explanation forthis:
Demonstration 2.3.
We express √ by its decimal fraction √ . · · · ,then . , . , . , . , . , · · · will become a candidate sequence of ratio-nal numbers. Once we have √ . · · · ∗ | {z } m − , , , · · · , | {z } n , ⋆, · · · , where ∗ , ⋆ ∈{ , , , , , , , , } and in this demonstration we can assume ∗ = 4 = ⋆ , then the ( m − − th number of this sequence should be . · · · | {z } m − , the ( m − i ) − th one . · · · | {z } m − · · · | {z } i , i ∈ { , · · · , n } and the ( m + n ) − th one . · · · ∗ | {z } m − · · · | {z } n ⋆ · · · .With the same rule, we can modify all the places several s come out one by one. Now we get a sequence of 2-resonant points { ω m } m ∈ N which approaches ω stepby step. It’s a slow but steady approximated process. Accordingly, we can find( a m , b m ) ∈ N such that ω m = ( a m l m , b m l m ). Between ω m and ω m +1 , along thesepartition lines, we can find a ω m, as a medium 2-resonant point (see figure 3),which can be expressed formally by ( a m l m , b m +1 / l m +1 ) or ( a m +1 / l m +1 , b m l m ), where a m +1 / , b m +1 / ∈ N . We could only consider the ω m +1 / of a former case in this paper.Finally, we can connect all these { ω m , ω m +1 / } m ∈ N along the Lattice lines andget one asymptotic resonant plan P ω = ∪ m ∈ N { Γ ωm } . We call Γ ωm . = { ω m Γ ωm, −→ ω m +1 / ωm, −→ ω m +1 } one-step transport process and show its several fine prop-erties. SYMPTOTIC TRAJECTORIES OF KAM TORUS 15
Figure 3.
Definition 2.4.
In our case of 2.5 degrees of freedom, time variable will be involvedin. Let ˜ ω . = ( ω, ∈ R be the frequency, Λ m . = span Z { ( l m , , − a m ) , (0 , l m , − b m ) } be the Lattice vertical to ˜ ω m and Λ m + be the one vertical to ˜ ω m +1 / . Thecorresponding maximal Lattices can be denoted by Λ maxm and Λ maxm +1 / . d m . = dist ( ω m , ω ) and d m +1 / the distance of ω m +1 / to ω .It’s obvious that d m +1 ≤ d m +1 / ≤ d m and d m ≤ √ l m . Recall that a m l m , b m +1 / l m +1 , a m +1 / l m +1 and b m l m may be reducible, so Λ m = span Z { ( l m , , − a m ) , (0 , l m , − b m ) } andΛ m +1 / = span Z { ( l m , , − a m ) , (0 , l m +1 , − b m +1 / ) } are unnecessarily maximal Lat-tices. Besides, we will face a new difficulty: there may be ‘stronger’ 2-resonantobstructions in Γ ωm , i.e. ω ∗ = ( a m l m , c m ι m ) ∈ Γ ωm with l m ≥ ι m , even l m ≫ ι m . Laterwe will transform these difficulties into several conditions of { f k } k ∈ Z and solvethem, with the aforementioned ‘Pickup’ and ‘Shear’ operators. Lemma 2.5. (1) Γ ωm,im ∈ N are line segments parallel but not collinear with eachother, i = 1 , . (2) Γ ωm ∩ Γ ωm +1 = ω m +1 , and Γ ωm ∩ Γ ωn = ∅ , here m ∈ N , n = m ± . (3) ω = ω ′ and they both locate on P ω , then Λ ω ∩ Λ ω ′ is either a one-dimensionalLattice, or (0 , , ∈ Z . The former case happens iff ω lies on the same Γ m,i with ω ′ , i = 1 , . The latter case happens iff they lie on different Γ m,i segments.Proof. We omit the proof here since these can be easily deduced from our construc-tion. (cid:3)
In the next, we will find the Stable Normal Forms of system (1.5) in different do-mains which are valid for all the resonant segments { Γ pm } m ∈ N with a KAM iterationapproach. Since h ( p ) is strictly positive definite, we get the Γ pm via a diffeomor-phism from Γ ωm . We just need to give the demonstration on Γ m, and other resonant † , CHONG-QING CHENG ‡ segments can be treated in the same way. This process can be operated for any m ∈ N , so we could assume m ≫ Stable Normal Form and unified expression of Hamiltoniansystems
First, we need to divide the Stable Normal Form into 2-resonance and 1-resonancetwo different cases as ω ∈ Γ ωm, is differently chosen.3.1. Except for the two end points ω m and ω m +1 / , there areinfinite many other 2-resonant points on Γ ωm, . We just need to consider finitelymany of them, which we call ‘sub 2-resonant’ points. In other words, we onlyconsider the 2-resonant case ω ∗ = ( a m l m , c m ι m ), and ι m could be chosen any integerbetween 1 and l m (1+ ξ ) , here ξ > H ( p, q, I, t ) = I + h ( p ) + p σ f ( q, t ) . = ˜ h ( p, I ) + p σ f ( q, t ) . We can formally give one step KAM iteration to this system: H + = H ◦ Φ = H + { H, W } + Z (1 − t ) {{ H, W } , W } ◦ φ t ( p, q ) dt = ˜ h + p σ f ( q, t ) + { ˜ h, W } + { p σ f ( q, t ) , W } + Z (1 − t ) {{ ˜ h, W } , W } ◦ φ t dt + h.o.t. Here Φ is an exact symplectic transformation defined in the domain B ( p ∗ , δ + ) × T × S of phase space, with W its Hamiltonian function and Φ = φ t =1 the time-1mapping. ∇ h ( p ∗ ) = ω ∗ and δ + ≤ δ , with δ as the available radius at first. Recallthat we could make m ≫ D h (0) . = A ∼ O (1) relative to m .We could take W = p σ g ( q, t ) formally and g ( q, t ) = T ∗ m R T ∗ m f ( q + ω ∗ t, t ) tdt .Here T ∗ m is the period of frequency ω ∗ and we know T ∗ m = lcm ( l m , ι m ) ≤ l m · ι m .Then we can solve the cohomology equation and get: H + = ˜ h ( p, I ) + p σ [ f ]( q, t ) + h ∆ ω, p σ ∇ θ g i + σp σ − ( f ∇ θ g − ∇ theta f g )+ Z (1 − t ) {{ ˜ h, W } , W } ◦ φ t dt + h.o.t Here θ . = ( q, t ) and ∆ ω . = ω − ω ∗ . Again we recall that p , θ , f are g all vectors.Besides, we have {{ ˜ h, W } , W } = ( W tp + h pp W q + ωW qp ) W q − ( W tq + ωW qq ) W p , of which only the value under k · k C , B norm we care.Recall that it’s just a formal derivation, so we need a list of conditions to makeit valid. SYMPTOTIC TRAJECTORIES OF KAM TORUS 17 • First, we should control the drift of action variable p , i.e. restricted in thedomain B ( p ∗ + , δ + ), the quantity of drift doesn’t exceed ∆ δ . = δ − δ + . Without lossof generality, we can assume R T f ( q, t ) dqdt = 0 and p ∗ + = p ∗ . Then we need k Z ∂W∂q ◦ φ t dt k B ( p ∗ ,δ + ) ≤ k W q k B ( p ∗ ,δ ) ≤ ∆ δ. As we know,(3.2) 1 l m ≤ d ∗ m ≤ √ l m , m ≫ . This is because the special resonant plan we choose. So we just need k p k σ k g q k B ( p ∗ ,δ ) ≤ ∆ δ . On the other side, k p k σ ≤ ( k p − p ∗ k + k p ∗ k ) σ ≤ σ X i =0 C iσ k p − p ∗ k i k p ∗ k σ − i , so we need(3.3) k p σ g q k B ( p ∗ ,δ ) ≤ c d ∗ σm T ∗ m k f k C , B ≤ δ , by taking ∆ δ = δ and δ ≤ d ∗ m . Here c is a constant depending on c and σ . Ac-tually, the estimation here is rather loose, owing to the robustness of KAM method. • Second, we must assure that the tail term R ( p, q, t ) and the resonant term Z ( p, q, t ) are strictly separated, i.e. k R k C ,δ + ≪ k Z k C ,δ + . Here Z ( p, q, t ) = p σ [ f ]( q, t ), R = R + R , R = ∆ ω · p σ g θ , and R = σp σ − ( f ∇ θ g − ∇ theta f g ) + Z (1 − t ) {{ ˜ h, W } , W } ◦ φ t dt + h.o.t. Actually, we have R ( p, q, t ) = d ∗ σ − m ˜ R ( q, t ) + h.o.t , and k ˜ R k C ≤ T ∗ m k f k C , k R k C ≤ δd ∗ σm T ∗ m k f k C . On the other side, the resonant term satisfies: Z = p σ [ f ] = p σ X ( k,l ) ⊥ ˜ ω ∗ ( k,l ) ∈ Z \{ } f k,l exp πi ( h k,q i + l · t ) , where ˜ ω ∗ = ( a m l m , c m ι m , c m ι m is irreducible, but a m l m may be not. There are twoaforementioned difficulties we should face:(1) a m l m may be reducible and a m l m = λa ′ m λl ′ m . l ′ m ≪ l m could even happen andmake the estimation of Z of m ambiguous.(2) the case ι m ≪ l m may happen. Later we will see that this may cause a big‘obstruction’ to the persistence of NHIC according to Z . † , CHONG-QING CHENG ‡ We denote the maximal Lattice vertical to ˜ ω ∗ by Λ maxω ∗ = span Z { ~e , ~e } , with ~e = ( l ′ m , , − a ′ m ) and ~e = (0 , ι m , − c m ). Then we can translate Z into: Z = p σ [ f ] = p σ [ f ] + p σ [ f ] = p σ X l ∈ Z f l~e exp πil ( l ′ m q − a ′ m t ) + p σ X l,k ∈ Z k =0 f l~e + k~e exp πi [ l ( l ′ m q − a ′ m t )+ k ( ι m q − c m t )] . We have ( l m , , − a m ) = λ~e , Λ λ~e . = span Z { λ~e } and Λ ω ∗ . = span Z { λ~e , ~e } . If(2) happens, [ f ] may be much larger than [ f ] by comparing their Fourier coeffi-cients. So we need f ( q, t ) ∈ B (0 , c ) to be well chosen and satisfies the followingconditions: C1: f ∈ B (0 , c ) satisfies: F ( f, Λ ω ∗ ) = F ( f, Λ maxω ∗ ), i.e. f k ≡ ∀ k ∈ Λ maxω ∗ \ Λ ω ∗ . C2: f ∈ B (0 , c ) satisfies: G ( f, Λ maxω ∗ , l m ) = F ( f, Λ maxω ∗ ), i.e. f k ≡ ∀ k ∈ Λ maxω ∗ and | k | ≤ l m .Since f ∈ B (0 , c ), we have | f k | ≤ c | k | r from Lemma 2.1. Here c = c ( r, c ) is aconstant. Later we also use a symbol ⋖ ( ⋗ ) to avoid too much c i constant involved,which means ≤ ( ≥ ) by timing a O (1) constant on the right side. These symbolsare firstly used by J. p¨oschel in [43].Based on the two Fourier conditions above, we will give the first uniform restric-tion on [ f ] . U1:
As a single-variable function of h λ~e , θ i , p ∗ σ [ f ] has a unique maximal valuepoint, at which it is strictly nondegenerate with an eigenvalue not less than c d ∗ σm l m ( r +2) ,since m ≥ M ≫
1. Here c ≥ Remark . This restriction assures the strength of normal hyperbolicity corre-sponding to the main direction of Γ m, . Its order of m is controllable and uniform.It’s a necessary demand to resist infinitely many cusp remove. Remark . We also recall that the index r + 2 in U1 can be replaced by any r + ζ ( ζ ≥ ζ will be involved in to assure | R | ≪ | Z | . Thestronger hyperbolicity is, the easier to assure the existence of NHICs. So we justconsider the case of r + 2. Remark . During the whole Γ m, except ω m +1 / , transformations between dif-ferent resonant lines are not involved in. We just need to construct a NHIC ‘tran-spierce’ the whole Γ m, , so we don’t give any restriction to [ f ] temporarily (seefigure 4). SYMPTOTIC TRAJECTORIES OF KAM TORUS 19
Figure 4.
Now let’s satisfy the two bullets above and finish this KAM iteration with: d ∗ σ − m T ∗ m ⋖ δ, (3.4) δT ∗ m ≪ d ∗ r +2 m , (3.5) δ ≤ d ∗ m , (3.6) m ≥ M ≫ . (3.7)We can sufficiently take(3.8) σ > r + 2and(3.9) δ ≪ l − m ( r +4+ ξ ) . Recall that σ can be chosen properly large from Lemma 1.7. Remark . In the above process, we left an index ξ > ξ is, the more ‘sub 2-resonant’ points we should consider.However, we hope the number of these points as little as possible, since the diffusionmechanism of 2-resonance is much complex than that of 1-resonance. In otherwords, we’ll apply 1-resonant mechanism along Γ m, as much as possible. Thatneeds an estimation of the lower-bound of ξ in the next subsection.3.2. We first revise several symbols which are valid only in thissubsection. As Figure 5 shows us, Γ m, is devided into several 1-resonant segmentsby sub 2-resonant points. Of each segment S we could find a tube-neighborhoodwith radius δ , on which we hope to get a similar Stable Normal Form by KAM it-erations. Let δ + be the radius of ball-neighborhood of 2-resonant points, for whichthe restriction (3.9) holds. In order to make the tube-neighborhoods approach 2-resonant points as near as possible, we will choose δ + as less as possible under thepremise that 2-resonant Stable Mornal Form is valid. † , CHONG-QING CHENG ‡ Figure 5.
We devide f ( q, t ) = P ( k,l ) ∈ Z f k,l exp i π ( h k,q i + lt ) into T K f and R K f , to expressthe partial sums of Fourier series of | ( k, l ) | ≤ K and | ( k, l ) | > K . From Lemma2.1 we know that k R K f k C ≤ κ K − r +6 k f k C r . Still we could give a formal KAMiteration in the tube-neighborhood of S : H + = H ◦ Φ = H + { H, W } + Z (1 − t ) {{ H, W } , W } ◦ φ t ( p, q ) dt = ˜ h + p σ f + { ˜ h, W } + { p σ f, W } + Z (1 − t ) {{ ˜ h, W } , W } ◦ φ t dt + h.o.t = ˜ h + p σ T l m (1+ ξ ) f + ˜ ωW θ + { p σ f, W } + p σ R l m (1+ ξ ) f + Z (1 − t ) {{ ˜ h, W } , W } ◦ φ t dt + h.o.t. Here Φ is an exact symplectic transformation defined in the domain T ( S , δ ) × T × S of phase space, with W its Hamiltionian and Φ = φ t =1 the time-1 mapping. Wedenote by R Λ . = { ω ∈ R |h k, ω i = 0 , ∀ k ∈ Λ ⊂ Z , ω ≡ } the set of frequenciesvertical to Λ-Lattice, then Γ m, ⊂ R Λ λ~e . Recall that we only consider the ω of which there exists a k ′ ∈ Z not in Λ ~e , such that k ′ ⊥ ω and | k ′ | ≤ l m (1+ ξ ) .We denote by R Λ ξλ~e , + the set of all these frequencies. Then we can solve thecohomology equation formally in the domain T (Γ m, , δ ) \ B ( R Λ ξλ~e , + ∩ Γ m, , δ + ) andget the resonant term Z ( q, t ) = p σ X ( k,l ) ∈ Λ λ~e | ( k,l ) |≤ l m (1+ ξ ) f k,l exp i π ( h k,q i + lt ) , owing to C1 and C2 conditions.To ensure this formal KAM iteration valid, we also face the two difficulties asthe bullet parts in previous subsection of 2-resonance. • First, we need to control the drift value of action variable p . Let’s sufficientlytake(3.10) k Z ∂W∂q ◦ φ t dt k C , δ ≤ k W q k C ,δ ≤ δ . SYMPTOTIC TRAJECTORIES OF KAM TORUS 21
Here we directly shrink the radius of tube-neighborhood of S to δ . Also we knowthat W = p σ X ( k,l ) / ∈ Λ λ~e | ( k,l ) |≤ l m (1+ ξ ) f k,l πi h ˜ ω, ( k, l ) i exp i π ( h k,q i + lt ) . • Second, we need k R k C ,δ ≪ k Z k C ,δ . As the case of 2-resonance, we need auniform condition of Z here: U2:
As a single-variable function of h λ~e , θ i , ∀ p ∈ S , − p σ [ T l m (1+ ξ ) f ] has aunique minimal value point, which is strictly nondegenerate with the eigenvaluenot less than c d σm l m ( r +2) ( m ≥ M ≫ c = c ( c , ξ ) ≥ is a new constant, and d m is the distance between p ∈ Γ pm, and p . Remark . Here U2 can be considered as a reinforcement of U1 , which helpsus avoid a complicated 1-resonant bifurcation problem which is faced in [16] and[8]. A special example satisfying U2 is to take p σ f ( q, t ) = ( p + p ) σ/ cos( h θ, λ~e i )with even σ . We can see that in this case NHIC corresponding to [ f ] does exist asa single connected cylinder without decatenation.Actually, we can loosen U2 to a general condition with bifurcation points (seeAppendix 8.2). To avoid the verbosity of narration, we still assume U2 as a properuniform restriction. Later, in the Appendix 8.2 we will deal with the bifurcatedcase use a genericity developed in [16]. Definition 3.6.
Let P Λ : R → R be the projection of a vector ω to the real-expanded space span R { Λ } and P Λ + be the projection to span R { Λ , k } , if there exists k ∈ Z not lie on Λ.In the 1-resonant situation we have Λ = Λ λ~e , P = P Λ λ~e and P + = P Λ λ~e , + . Wedon’t care the concrete form of k , but | k | ≤ l m (1+ ξ ) is demanded in the followingestimation. ∀ ˜ ω ∈ T ( R Λ λ~e ∩ Γ m, , δ ) \ B ( R Λ ξλ~e , + , δ + ), we have: h k, ˜ ω i = h k, P + ˜ ω i = h Qk, Q ◦ P + ˜ ω i + h P k, P ◦ P + ˜ ω i = h Qk, ( P + − P )˜ ω i + h P k, P ˜ ω i , with Q = Id − P , then |h k, ˜ ω i| ≥ |h Qk, ( P + − P )˜ ω i| − |h P k, P ˜ ω i| . Notice that Qk is parallel to ( P + − P )˜ ω and P k is parallel to P ˜ ω . We actually get |h k, ˜ ω i| ≥ | Qk || ( P + − P )˜ ω | − | P k || P ˜ ω | , and furthermore(3.11) |h k, ˜ ω i| ≥ q δ − δ p l m + a m − l m (1+ ξ ) δ. We can write the right side of above inequality by α . That’s the so called ‘smalldenominator’ problem, so we need α > Remark . This estimate of h k, ω i was firstly given in [44]. Here is just a directapplication of that. † , CHONG-QING CHENG ‡ On the other side, we can estimate the tail term R by:(3.12) k R k C = k p σ R l m (1+ ξ ) f k C ⋖ κ d σm k f k C l m (1+ ξ )( r − , If we take R . = { p σ f, W } + R (1 − t ) {{ H, W } , W } ◦ φ t ( p, q ) dt + h.o.t , then we have(3.13) k R k C ≤ d σ − m k f k C α l m (1+ ξ ) . Recall that d m is the distance between p ∈ S and p , and formula (3.2) is stillvalid. Based on U2 , we need the followings to ensure the KAM iteration valid:(1) k R k C ≪ k Z k C ⇒ ( r − ξ ) > r + 2 ⇒ ξ > r − ,(2) k R k C ≪ k Z k C ⇒ d σ − m α ≪ l − m ( r +4+2 ξ ) ,(3) d σm α l m (1+ ξ ) ⋖ δ (drift value control).We can roughly take α ∼ O ( δ + l m ). Recall that δ + ≪ l − m ( r +4+ ξ ) from (3.9) and d m ∼ O ( l − m ), then we have: δ − l m (2+ ξ ) d σm ⋖ δ,δ + ≫ d σ − m l m (6+ r +2 ξ ) ,δ · l m (2+ ξ ) ⋖ δ + ,ξ > r − . These can be further transformed into(3.14) l − m ( σ − − r − ξ )2 ≪ δ + ≪ l − m ( r +4+ ξ ) , (3.15) l − m ( σ − − ξ ) δ + ⋖ δ ⋖ l − m (2+ ξ ) δ + , (3.16) ξ > r − . So we need the following index inequalities: r + 4 + ξ <
12 ( σ − − r − ξ ) , (3.17) ξ > r − . (3.18)A new index restriction of(3.19) σ > r + 4 ξ + 15will replace formula (3.8).Here we give a lower bound for ξ > r − . As is known from the previous sub-section, we’ll take ξ as small as possible, actually ξ = r − is enough. We can seethat ξ → r → ∞ . On the other side, we know the strict lower bound of δ + is l − m ( σ − − r − ξ )2 , and k Z k C ∼ O ( l − m ( σ + r +2) ). So we can roughly estimate the orderrelationship by(3.20) inf δ + ∼ O ( k Z k σ − − r − ξ σ + r +2) C ) . SYMPTOTIC TRAJECTORIES OF KAM TORUS 23
We can see that the index in the above formula tends to as σ → + ∞ . Laterwe’ll see that the index ‘ ’ plays a key role in the ‘homogenized’ method, whichwas firstly used in [16] and [34]. Nonetheless, we can take σ properly large suchthat the index of (3.20) greater than . Similar estimation was also obtained in[16] and [8], with an ( ǫ, δ )-language. Remark . This approach of Stable Normal Form was firstly developed by LochakP. and P¨oschel J. in [33] and [44], in solving a Nekhoroshev estimation problem.Notice that we can get a better estimation with more steps of KAM iterations, butwe will face some new difficulties: the loss of regularity and non-linearity of opera-tors F and G about f . To avoid the technical verbosity, only one-step iteration isoperated in this paper. This is enough for our construction and makes the wholeproof easy to read.Anyway, we can get Stable Normal Forms for both 1-resonance and 2-resonance,in the domain covering the whole Γ pm, × T × S . Similarly, we can repeat thisprocess for Γ m, × T × S , and then the whole resonant plan P ω . During thisprocess, new versions of U1 and U2 can be raised on Γ m, parallelly.3.3. Canonical coordinate transformations for Stable Normal Forms.
Re-call that the resonant term Z ( p, q, t ) of the Stable Normal Form is resonant withrespect to ω , the current frequency, so we have Z = Z ( p, h ~e , θ i ) (1-resonance)or Z = Z ( p, h ~e , θ i , h ~e , θ i ) (2-resonance). So we can transform the correspondingStable Normal Form into a canonical form which is universal for the whole P ω . • We know the Stable Normal Form of this case is(3.21) H = h ( p ) + I + p σ ([ f ] + [ f ] ) + R ( p, q, t ) , on B ( p ∗ , δ ) × R × T × S . Here [ f ] only depends on h λ~e , θ i , and [ f ] depends on h λ~e , θ i and h ~e , θ i . Recallthat ι m varies from 1 to l m (1+ ξ ) , which brings some difficulties to our canoni-cal transformation. Actually, this canonical transformation is a linear symplecticmatrix, so we need the following condition to make the elements of matrix homo-geneous. C2’ : If µ = min k ∈ Z + {| k~e | > l m } and Λ λ,µω ∗ . = span Z { λ~e , µ~e } , we take f ∈ B (0 , c ) satisfying F ( f, Λ ω ∗ ) = F ( f, Λ λ,µω ∗ ).Let ~e = (0 , , µι m l m ) and(3.22) Ξ . = ( λ~e , µ~e , ~e ) t = l m − a m µι m − µc m µι m l m × be a unimodular matrix. We can get a symplectic transformation via:(3.23) (cid:18) xs (cid:19) = Ξ (cid:18) qt (cid:19) , (cid:18) pI (cid:19) = Ξ t (cid:18) yJ (cid:19) + (cid:18) p ∗ (cid:19) . † , CHONG-QING CHENG ‡ Figure 6.
Under this transformation, we can change system (3.21) into(3.24) H = Jµι m l m + h ′ ( y , y ) + p ∗ σm [ f ] ( x ) + p ∗ σm [ f ] ( x , x ) + R ′ ( x, y, s ) , with ( y , y ) ∈ [ − δl m , δl m ] × [ − δµι m , δµι m ]. Here we move the higher order terms of Z into the tail term and get a new R ′ . Besides, we have k R ′ k C ⋖ O ( δd σ − m T ∗ m ) , and h ′ ( y ) = h ′ (0) + ∇ h ′ (0) y + 12! D h ′ (0) y + 13! D h ′ (0) · · · Witout loss of generality, we can assume h ′ (0) = 0. We also have ∇ h ′ (0) = 0 ofthis formula, and(3.25) D n h ′ (0) = h Θ , D n h ( p ∗ ) Θ t i , Θ t i , · · · , Θ t i | {z } n − , where Θ = (cid:18) l m µι m (cid:19) is an amplified matrix. Remark . In the later paragraph, we often apply the rescaled system H ′ = µι m l m H which is more convenient. • In the same way, we know the Stable Normal Form of this case is(3.26) H = h ( p ) + I + p σ [ f ] ( h λ~e , θ i ) + R ( p, q, t ) , where ( p, I, q, t ) ∈ T ( R Λ λ~e , δ ) \ B ( R Λ λ~e , + , δ + ) × R × T × S . For each segment S between two 2-resonant points, we can find a finite sequence of open balls to coverit, i.e. S ⊂ {B ( p ∗ i , δ ) } N m i =1 (see Figure 6). Recall that ω ∗ i is a 1-resonant frequencycorresponding to p ∗ i . In the domain B ( p ∗ i , δ ) × R × T × S we can find a similarlinear symplectic transformation with(3.27) Ξ = l m − a m l m
00 0 l m , SYMPTOTIC TRAJECTORIES OF KAM TORUS 25 and(3.28) (cid:18) xt (cid:19) = Ξ (cid:18) qt (cid:19) , (cid:18) pI (cid:19) = Ξ t (cid:18) yJ (cid:19) + (cid:18) p ∗ i (cid:19) . Based on this transformation, system (3.26) will become:(3.29) H = Jl m + h ′ ( y, p ∗ i ) + p ∗ σi [ f ] ( x ) + R ′ ( x, y, s ) , where y ∈ B (0 , δl m ). Here p ∗ i is a parameter to mark the ball neighborhoods inwhich we apply the transformation. Also we throw the higher order terms of Z intotail terms and get a new R ′ . Notice that ∇ h ′ (0 , p ∗ i ) = (0 , l m ω ∗ i, ) , ω ∗ i = ( ω ∗ i, , ω ∗ i, ) , and D n h ′ (0 , p ∗ i ) = l nm D n h ( p ∗ i ) . Remark . From system (3.29), we can see that h ′ ( y, p ∗ i ) and Z ( x , p ∗ i ) dependon a parameter variable p ∗ i . Later, in the second part of Appendix we will deal withthe bifurcation cases with a genericity of [16].3.4. Transition from 1-resonance to 2-resonance.
Along Γ m, , we need to construct diffusion orbits with the frequency changingfrom ω m to ω m +1 / . As ω m +1 / is a typical 2-resonant point, once the diffusionorbits pass it, we can repeat this process along Γ m, because U1 and U2 are stillvalid for it. So the key to achieve this process is to overpass ω m, / . Once onestep transport process Γ m is finished, all the transition plan P ω could be overcomebecause of our self-similar structure.Recall that there still exists finitely many ‘sub 2-resonant’ points at Γ m, to beoverpassed. Different from ω m +1 / , there’s no transitions between resonant lines atthese points. So our plan is to find a NHIC with ( x , y ) as ‘fast-variables’ (in arough sense) to overcome these points which persist under small perturbations (seeFigure 7).To achieve these, we need to weaken the hyperbolicity corresponding to fast vari-ables ( x , y ) for the sub 2-resonant points, and create a proper domain in whichdifferent NHICs can be connected with each other for ω m +1 / . More uniform con-ditions and a ‘weak-coupled’ mechanism will be involved in this section.From (3.20) we know, the index is contained in [ , ). So there must be aoverlapping domain in which both the Stable Normal Forms of 1-resonance and of2-resonance valid. We can carry out one-step KAM iteration again in a domain( B ( p ∗ m , δ + ) \ B ( p ∗ m , K k Z k C )) ∩ T (Γ pm, , k Z k C ) (see Figure 8). Here K ≫ H ′ system correspondingto (3.24) can be rewritten as: H ′ = J + µι m l m ( h ′ ( y ) + p ∗ σm [ f ] ( x ) + p ∗ σm [ f ] ( x , x ) + R ′ ( x, y, t )) , † , CHONG-QING CHENG ‡ Figure 7. with y ∈ [ − δ + l m , δ + l m ] × [ − δ + µι m , δ + µι m ]. We can divide [ f ] into:[ f ] ( x ) = X l,k ∈ Z k =0 f lλ~e + kµ~e exp πi [ lx + kx ] = X k ∈ Z k =0 f kµ~e exp πikx + X l,k ∈ Z k,l =0 f lλ~e + kµ~e exp πi [ lx + kx ] . = [ f ] , ( x ) + [ f ] , ( x , x )and raise a new uniform condition: U3:
As a single-variable function, p ∗ σm [ f ] , ( x ) has a unique maximal point(without loss of generality we assume this point by x = 0) at which [ f ] , is non-degenerate. Besides, k p ∗ σm [ f ] , k C ≤ c d ∗ σm ( µι m ) r +2 with ≤ c < c for m ≥ M ≫ Remark . This condition is aiming to weaken the hyperbolicity according to( x , y ) variables. Recall that when p ∗ m = p m +1 / , also U1 should be satisfied andthat’s why a comparison of c and c is involved. U4: If k p ∗ σm [ f ] , k C ∼ O ( d ∗ σm ( l m ) r +2+ η ) with η ≥
0, we restrict that k p ∗ σm [ f ] , k C ⋖ L ( l m ) r +2+ η , L ≫ m ≥ M ≫ L ≫ SYMPTOTIC TRAJECTORIES OF KAM TORUS 27
Figure 8.
Remark . This is the so-called ‘weak-coupled’ mechanism. We actually weakenthe coupled Fourier coefficients to make the system (3.24) more like a nearly inte-grable system at a proper domain of 2-resonance (later in the section of homoge-nization we will see that). Notice that here a sufficiently large number L is involved,we will give a priori estimation of L then take m sufficiently large comparing to itin the following proof.In the domain ( B ( p ∗ m , δ + ) \ B ( p ∗ m , K k Z k C )) ∩ T (Γ pm, , k Z k C ), we can carry outone more step KAM iteration for the system (3.24): H ′ + = H ′ ◦ Φ = H ′ + { H ′ , W } + Z (1 − t ) {{ H ′ , W } , W } ◦ φ t ( x, y ) dt = J + µι m l m ( h ′ + p ∗ σm [ f ] + p ∗ σm [ f ] ) + µι m l m { h ′ , W } + R ′ , + , where the new tail term R ′ , + satisfying: R ′ , + = µι m l m ( R ′ + { p ∗ σm [ f ] + p ∗ σm [ f ] + R ′ , W } ) + Z {{ H ′ , W } , W } ◦ φ t ( x, y ) dt. Here the cohomology equation to solve is p ∗ σm [ f ] + { h ′ , W } = p ∗ σm [[ f ] ]( x ) + ∆ ωW x , † , CHONG-QING CHENG ‡ with ∆ ω = ω − ω ∗ , ω ∗ = ∇ h ′ ( y ∗ ) = (0 , ω ∗ ) and y ∗ ∈ Γ ym, ∩ ( B ( p ∗ m , δ + ) \B ( p ∗ m , K k Z k C )). So we can formally take W = 1 T ∗ Z T ∗ p ∗ σm [ f ] ( x , x + ω ∗ s ) sds and p ∗ σm [[ f ] ]( x ) = 1 T ∗ Z T ∗ p ∗ σm [ f ] ( x , x + ω ∗ s ) ds, where we denote by T ∗ = ω ∗ .Since this iteration is operated in B ( y ∗ , δl m ) × B ( y ∗ , δµι m ), we need to considerthe drift value of y variables. Recall that µι m ≥ l m , so we can sufficiently take:(3.30) k Z ∂W∂x ◦ φ t dt k B ( y ∗ , δ ) × T ≤ k W x k B ( y ∗ ,δ ) ≤ δ µι m . On the other side, we have: k W x k B ( y ∗ ,δ ) ≤ T ∗ k p ∗ σm [ f ] k C , B ⋖ d ∗ σm ( µι m ) r +2+ η ω ∗ (from U4 ) ⋖ K · d ∗ σm ( µι m ) r +2+ η · µι m k Z k (from U2 and (3.20)) ⋖ δ K µι m ≪ δ µι m , K a posteriori sufficiently largewith δ = k Z k . Besides, the new tail term R ′ + . = R ′ . + + R ′ , + = ∆ ωW x + R ′ , + , and we have k R ′ + k C ⋖ µι m l m ( T ∗ µι m δd ∗ σm µι m ) r +2+ η ) ⋖ µι m l m k Z k K · l m ( r +2) ( µι m ) r +2+ η , (3.31)during which the largest term is ∆ ωW x .Notice that the new resonant term p ∗ σm [[ f ] ]( x ) won’t influence the main valueof p ∗ σm [ f ] ( x ) because of U3 condition. Actually we can first take L properly largesuch that L − ≪ c i , i = 1 , , , · · · ,
6, then take m ≥ M ≫ L accordingly. Butthe difference between Z and R ′ + is just limited by a multiplier K − (see inequality(3.31)). Later we will see that the NHICs in this domain present some kind of‘crumpled’ form.3.5. Homogenized system of 2-resonance.
After passing the transition partsfrom 1-resonance to 2-resonance, now we reach a domain ( y , y ) ∈ B (0 , O ( K δl m )) ×B (0 , O ( K δµι m )), with δ = k Z k / C . Now we can homogenize system (3.24) into a clas-sical mechanical system with small perturbation, which benefits us with many fineproperties. We will see that in the next section. SYMPTOTIC TRAJECTORIES OF KAM TORUS 29
For convenience, we still rewrite system (3.24) here: H ′ = J + µι m l m ( h ′ ( y ) + p ∗ σm [ f ] ( x ) + p ∗ σm [ f ] ( x , x ) + R ′ ( x, y, t )) , where ( y , y ) ∈ B (0 , O ( K δl m )) × B (0 , O ( K δµι m )). We can transform this system with arescale symplectic transformation, via:(3.32) x x sy y J = l m µι m δl m µι m δl m δµι m
00 0 0 0 0 δ l m µι m · X X SY Y e , where δ ∼ O ( d ∗ σ/ m l m ( r +2) / ). These new variables will satisfy a new O.D.E equationwith a different proportion:(3.33) X ′ . = ∂X∂S = ∂X∂s · δl m µι m = 1 δl m µι m l m µι m ! ∂x ∂s∂x ∂s ! . Since we have(3.34) ∂x ∂s∂x ∂s ! = l m µι m ˙ x ˙ x ! = l m µι m " ∂h ′ ∂y ∂h ′ ∂y ! + ∂R ′ ∂y ∂R ′ ∂y ! , we can throw O ( y ) term of h ′ into tail and get: X ′ = D h ( p ∗ m ) Y + 1 δ Θ − × ( O ( y ) + ∇ y R ′ )= D h ( p ∗ m ) Y + O ( δY ) + 1 δ ∇ Y R ′ . (3.35)On the other hand, we have:(3.36) Y ′ . = ∂Y∂S = ∂Y∂s · δl m µι m = 1 δl m µι m l m δ µι m l m ! ∂y ∂s∂y ∂s ! , (3.37) ∂y ∂s∂y ∂s ! = l m µι m ˙ y ˙ y ! = − l m µι m " ∂Z ′ ∂x ∂Z ′ ∂x ! + ∂R ′ ∂x ∂R ′ ∂x ! with Z ′ ( x ) . = p ∗ σm [ f ] ( x ) + p ∗ σm [ f ] ( x , x ), then we get(3.38) Y ′ = − δ ∇ X Z ′ ( X , X ) − δ ∇ X R ′ . From (3.35) and (3.38) we can rescale H ′ into a new system of variables ( X, S, Y, e )(3.39) ˜ H = 12 h Y t , D h ( p ∗ m ) Y i + ˜ Z ( X ) + ˜ R ( X, Y, S ) , where ˜ Z ( X ) = Z ′ δ and ˜ R ( X, Y, S ) = R ′ δ + O ( δY ). Actually, ˜ Z ( X ) can be devidedinto: ˜ Z ( X ) = ˜ Z ( X ) + ( l m µι m ) r +2 ˜ Z ( X ) + 1 L ( l m µι m ) r +2 ˜ Z ( X , X ) , † , CHONG-QING CHENG ‡ where k ˜ Z i k C ∼ O (1), i = 1 , ,
3. Recall that l m ≤ µι m ≤ l m (1+ ξ ) , so the ‘hardest’case to be considered is of a form(3.40) ˜ Z ( X ) = ˜ Z ( X ) + ˜ Z ( X ) + 1 L ˜ Z ( X , X ) . On the other side, the new tail terms k ˜ R k C ⋖ K d ∗ σ/ m l m ( r +6+2 ξ ) / . It’s sufficientlysmall comparing with ˜ Z as m ≫ Remark . This homogenization actually rectifies the ‘stretched’ effect of pre-vious canonical transformation of ( x, y ) variables and recover the phase space intoa normal O (1) − scale. Recall that when ω ∗ m is a sub 2-resonant point, there isno transformation between different resonant lines and we just need to prove thepersistence of NHIC with ( x , y ) as slow-variables (see Figure 7). So we just needto prove that for the case ω ∗ m = ω m +1 / , which corresponds to the hardest ˜ Z case(3.40). 4. Existence of NHICs and Location of Aubry sets
After the previous conditions and uniform restrictions been satisfied, we canprove the persistence of NHICs corresponding to different Stable Normal Forms indifferent domains of the phase space. We can divide them into 3 cases of differentmechanisms to deal with: 1-resonance, transition from 1-resonance to 2-resonanceand 2-resonance. Since our construction is self-similar, we just need to prove thatfor Γ m, and get the same persistence for that whole P ω . It’s remarkable that thehomogenization does help us greatly in the latter two cases.Actually, we will prove the persistence of ‘weak-invariant’ NHICs in proper do-mains where KAM iterations work. Here ‘weak-invariant’ means the vector field isjust tangent at each point of the cylinder, but unnecessarily vanished at the bound-ary. We call a NHIC ‘strong-invariant’ if it contains the whole flow of each points.These conceptions were firstly used by Bernard P. in [7].In the following we will firstly prove the former 2-cases with the skill used in [7],then prove the 2-resonance case with the help of the method developed in [16] andour special ‘weak-coupled’ construction.4.1. the persistence of wNHICs for 1-resonance. From (3.29) we know the canonical system for 1-resonance can be rewritten as: H = J + l m ( h ′ ( y, p ∗ i ) + p ∗ σi [ f ] ( x ) + R ′ ( x, y, t )) , y ∈ B (0 , δl m ) . The following holds:
SYMPTOTIC TRAJECTORIES OF KAM TORUS 31
Lemma 4.1.
There exists a modified system b H = J + l m [ h ′ ( y ) + 12 B ( y )( y − Y ( y )) + 12 p ∗ σi [ f ] ′′ (0) x ]+ l m [ 13! S ( y )( y − Y ( y )) χ ( l m ( y − Y ( y )) δ̟ ) + 13! p ∗ σi [ f ] ′′′ ( x ) x χ ( x ̺ )+ y ˜ R ′ χ ( l m yδ )] , with the assumption that x = 0 is the unique maximal point of p ∗ σi [ f ] ( x ) and χ ( · ) is a compactly supported smooth function which equals to on the unit ball B (0 , and outside of B (0 , . Here ( Y ( y ) , y ) with k y k ≤ δl m satisfies ∂h ′ ∂y ( Y , y , p ∗ i ) =0 , B ( y ) = ∂ h ′ ( Y ( y ) , y ) and S ( y ) is a 3-linear form on R depending smoothlyon y ∈ R . The value of ̟ ≪ ̺ ≪ H coincides with H on the domain {k x k ≤ ̺, k y − Y ( y ) k ≤ δ̟l m , x ∈ T , k y k ≤ δl m } , and coincides with the integrable system H = J + l m [ h ′ ( y ) + 12 B ( y )( y − Y ( y )) + 12 p ∗ σi [ f ] ′′ (0) x ]on the domain {k x k ≥ ̺, k y − Y k ≥ δ̟l m , x ∈ T , k y k ≤ δl m } . Proof.
We just need to expand the system H at the point { x = 0 , y = Y ( y ) , k y k ≤ δl m } into a finite Taylor series and then smooth it by multiplying a compactly sup-ported bump function. Recall that R ′ = y b R ′ + h.o.t. is the new tail term from(3.29). (cid:3) First, we can show that { (0 , x , Y ( y ) , y ) (cid:12)(cid:12) x ∈ T , k y k ≤ δl m } ⊂ T ∗ T is aNHIC according to system H . In order to simplify the corresponding equations, weset h ( y ) = h ′ ( y ) + 12 B ( y )( y − Y ( y )) , then we have ˙ x ˙ x ˙ y ˙ y = 1 l m ∂H∂y ∂H∂y − ∂H∂x − ∂H∂x = ∂ y h + B ( y ) 00 0 0 0 − p ∗ σi [ f ] ′′ (0) 0 0 00 0 0 0 · x x y − Y ( y ) y . We also get the eigenvalues p − B ( y ) p ∗ σi [ f ] ′′ (0), − p − B ( y ) p ∗ σi [ f ] ′′ (0), 0 and 0 forthe linear matrix above, with the corresponding eigenvectors( p B ( y ) , , q − p ∗ σi [ f ] ′′ (0) , t , ( − p B ( y ) , , q − p ∗ σi [ f ] ′′ (0) , t , (0 , , , t , † , CHONG-QING CHENG ‡ and (0 , , , t . The existence of NHIC according to H can be easily proved.Second, we set b R = 13! S ( y )( y − Y ( y )) χ ( l m ( y − Y ( y )) δ̟ )+ 13! p ∗ σi [ f ] ′′′ ( x ) x χ ( x ̺ )+ y ˜ R ′ χ ( l m yδ ) , then the equation corresponding to ˜ H can be written by ˙ x ˙ x ˙ y ˙ y = 1 l m ∂ ˜ H∂y ∂ ˜ H∂y − ∂ ˜ H∂x − ∂ ˜ H∂x = 1 l m ∂H∂y ∂H∂y − ∂H∂x − ∂H∂x + 1 l m ∂ b R∂y ∂ b R∂y − ∂ b R∂x − ∂ b R∂x . To get the persistence of NHIC corresponding to ˜ H , we need to control the value of b R under the norm k · k C , B and make it strictly separated from the spectrum radius p − B ( y ) p ∗ σi [ f ] ′′ (0), based on the classical theory of NHIC in [27]. So we get k S ( y ) k δ̟l m ≪ q − B ( y ) p ∗ σi [ f ] ′′ (0) ⇒ l m δ̟ ≪ r d ∗ σi l mr ,d ∗ σi l m ( r +2) ̺ ≪ q − B ( y ) p ∗ σi [ f ] ′′ (0) ⇒ ̺ ∼ O (1) small , δ d ∗ σ − i l mr ≪ q − B ( y ) p ∗ σi [ f ] ′′ (0) ⇒ d ∗ σ/ − i l mr/ ≪ δ,d ∗ σi l m (1+ ξ )( r − l m δ ≪ q − B ( y ) p ∗ σi [ f ] ′′ (0) ⇒ d ∗ σ i l m ( r +2 ξr − − ξ )4 ≪ δ, from (3.12), (3.13), (3.14) and (3.15). Then we can get the strict lower bound of δ by d ∗ σ i l m ( r +2 ξr − − ξ )4 . Once the aforementioned inequalities satisfied, we actuallyverify the persistence of NHIC for ˜ H in the domain: {k x k ≤ ̺, k y − Y k ≤ δ̟l m , x ∈ T , k y k ≤ δl m } . On the other side, ˜ H coincides with H in the domain: {k x k ≤ ̺, k y − Y ( y ) k ≤ δ̟l m , x ∈ T , k y k ≤ δl m } , So the cylinder is also ‘weak-invariant’ for H in the sense that at the two endsthere may be overflow (or interflow). Besides, we can see that the wNHIC (weakNormally Hyperbolic Invariant Cylinder) is of the form { ( t, x ( x , y , t ) , x , y ( x , y , t ) , y ) (cid:12)(cid:12) t ∈ S } , where we have k y ( x , y , t ) − Y ( y ) k ⋖ p ∗ σ + r +1 i B ( y )and k x ( x , y , t ) k ⋖ ̺ . SYMPTOTIC TRAJECTORIES OF KAM TORUS 33
Figure 9.
By changing p ∗ i along the Γ pm, , the ‘short’ wNHICs under different canonical coordi-nations can be joint into a ‘long’ wNHIC with only two ends overflow (or interflow).Notice that we take the lower bound of δ into (3.15) and get a restriction for δ + as well. As d i ∼ O (1), we can get an index estimation replacing (3.20):(4.1) d ∗ σ i l m (24+16 ξ − r − ξr )4 ∼ O ( k Z k σ +(1+2 ξ ) r − − ξ σ + r +2) ) . Here the index tends to 1/4 as σ → ∞ . This implies that the wNHIC of a 1-resonant mechanism can be expanded into the place at least O ( k Z k / ) approachingthe 2-resonant points, as long as σ is chosen properly large (see figure 8).4.2. the persistence of wNHICs for the transition part from 1-resonanceto 2-resonance. In this subsection, we will further expand the wNHIC got in the previous sub-section into the places O ( K k Z k ) approaching 2-resonant points. Different fromthe wNHIC of 1-resonant mechanism, here we will use the methods developed insubsection 3.4 and 3.5, i.e. both the twice KAM iteration and the homogenizationare involved.As Figure 9 shows us, we can pick up finitely many y i points on the Γ ym, fromindex ‘1/6’ to ‘1/2’ domains of 2-resonant points p ∗ m . Recall that system (3.24)is valid, and we can carry out twice KAM iteration in the domain B ( y i, , δl m ) ×B ( y i, , δµι m ), where y i = ( y i, , y i, ) and δ = k Z k / . We rewrite the system as:(4.2) H ′ = J + µι m l m ( h ′ ( y, y i ) + p ∗ σm [ f ] ′ ( x ) + R ′ ( x, y, y i , t )) , where U1 → restrictions are available. Here the parameter y i reminds us in whichdomain we carry out the twice KAM iteration. Recall that the estimation (3.31) † , CHONG-QING CHENG ‡ is still valid for this system 4.2 and we have k R ′ k ⋖ k Z k K for all chosen y i . Thiscomparison of the nearly same order failed to ensure the persistence of a commonwNHIC and we have to seek for other mechanisms to ensure the persistence of that.Homogenization will be involved to find the so-called ‘crumpled’ wNHIC (this def-inition was firstly proposed in [8]).With the same idea as subsection 3.5, we can ‘rescale’ system 4.2 via: x = l m X , x = µι m X , y − y i, = δl m Y , y − y i, = δµι m Y , s = Sδµι m l m . Then system (4.2) will become:˜ H = ω i, µι m δ Y + 1 / h Y t , D h ( p i ) Y i + ˜ Z ( X ) + ˜ R ( X, Y, S ) K , where p i is the corresponding point of y i and ∇ h ′ ( y i ) = (0 , ω i, ). Recall that the tailterm takes the largest value O ( K ) when y i ∈ B (0 , O ( K δl m )) ×B (0 , O ( K δµι m )), so we writeit down in a form ˜ R ( X,Y,S ) K . After this rescale we actually have Y ∈ B (0 , c ) ⊂ R , k ˜ Z k C ∼ O (1) and k ˜ R k c ∼ O (1). Here c ∼ O (1) is a constant.Under the new coordination, we have the following modified system: b H = ω i, µι m δ Y + ( Y ( Y ) , Y ) t · D h ( p i ) · (cid:18) Y ( Y ) Y (cid:19) + a i ( Y − Y ( Y )) + ˜ Z ′′ (0) X + χ ( X ̺ ) ˜ Z ′′′ ( X ) X + K χ ( Y − Y ( Y ) ̟ , Y c ) ˜ R ( X, Y, S ) , where χ ( · ) is a smooth bump function as it’s given in the previous subsection andwe formally denote by D h ( p i ) = (cid:18) a i b i b i c i (cid:19) . We can see that b H coincides with ˜ H in the domain: {k X k ≤ ̺, k Y − Y ( Y ) k ≤ ̟, k Y k ≤ c } . For convenience we can assume: b R = 13! χ ( X ̺ ) ˜ Z ′′′ ( X ) X + 1 K χ ( Y − Y ( Y ) ̟ , Y c ) ˜ R ( X, Y, S ) , and h ( Y ) = ω i, µι m δ Y + 12 ( Y ( Y ) , Y ) t · D h ( p i ) · (cid:18) Y ( Y ) Y (cid:19) + 12 a i ( Y − Y ( Y )) , where Y ( Y ) satisfies: ∂ Y h ( Y ( Y ) , Y ) = 0. Then we write down the correspond-ing equations as: X ′ = ∂ b H∂Y = a i ( Y − Y ( Y )) + ∂ b R∂Y ,Y ′ = − ∂ b H∂X = − ˜ Z ′′ (0) X − ∂ b R∂X ,X ′ = ∂ b H∂Y = ∂ Y h + ∂ b R∂Y , SYMPTOTIC TRAJECTORIES OF KAM TORUS 35 Y ′ = − ∂ b H∂X = − ∂ b R∂X . Also we can show that { (0 , X , Y ( Y ) , Y ) (cid:12)(cid:12) X ∈ T , k Y k ≤ c } is a NHIC corre-sponding to system H = h ( Y ) + 12 ˜ Z ′′ (0) X . Recall that the eigenvalues of the linear system H is ± q − a i ˜ Z ′′ (0) and 0 (2 order).To prove the persistence of NHIC for system b H , we need to control the value of b R under the norm k · k C , B . So the undetermined variables ̺ , ̟ should satisfy thefollowing: 13! ̺ ≪ q − a i ˜ Z ′′ (0) , K ̟ ≪ q − a i ˜ Z ′′ (0) , where we take ̟ ≤ c for convenience. Since q − a i ˜ Z ′′ (0) ∼ O (1) and K can bechosen properly large, we roughly take ̺ = K / and ̟ = K / and verify thepersistence of NHIC for b H in the domain: {k X k ≤ ̺, k Y − Y ( Y ) k ≤ ̟, k Y k ≤ c } . On the other side, b H coincides with ˜ H in the domain: {k X k ≤ ̺, k Y − Y ( Y ) k ≤ ̟, k Y k ≤ c } , so we actually proved the persistence of wNHIC for ˜ H in this domain which is ofthe form(4.3) { ( S, X ( X , Y , S ) , Y ( X , Y , S ) , X , Y ) } . We can see that k Y ( X , Y , S ) − Y ( Y ) k ⋖ K / → , and k X ( X , Y , S ) k ⋖ K → . uniformly as K → ∞ . But we can see that in the original coordination k ∂x ( x , x , s ) ∂y k ∼ O ( µι m l m K δ ) , the right side of which is quite large. That’s the meaning of ‘crumpled’ and we cansee that the nearer y i approaches 0, the more violent the crumple is. O ( µι m l m K δ ) isactually the largest estimation of crumple for y i ∈ B (0 , O ( K δl m )) × B (0 , O ( K δµι m )) (seeFigure 10). † , CHONG-QING CHENG ‡ Figure 10. the persistence of wNHICs for the 2-resonance ( ω m, ). After the homogenization of subsection 3.5, now we consider the following me-chanical system with the ‘hardest’ case potential function of (3.40):(4.4) H ( X, Y, S ) = H ( X, Y ) + ǫR ( X, Y, S ) , (4.5) H ( X, Y ) = 12 h Y t , D h ( p ∗ m ) Y i + Z ( X ) + Z ( X ) + εZ ( X , X ) , where ε = L and ǫ = K d ∗ σ/ m l m ( r +6+2 ξ ) / .Recall that the variable ( X , Y ) corresponding to the resonant line Γ m, and( X , Y ) corresponding to Γ m, when p ∗ m = p m, / (We only need to considerthis case since there’s no transition between resonant lines at other sub 2-resonantpoints). To complete our ‘weak-coupled’ structure, another condition is needed: C3: D h (0) = Id × . Remark . Once C3 is satisfied, we have D h ( p ∗ m ) = Id + O ( l m ) for sufficientlylarge m ≫
1. Then the ‘model system’ (4.5) is actually of a form:(4.6) H ( X, Y ) = 12 h Y t , Y i + Z ( X ) + Z ( X ) + εZ ( X , X ) + O ( k Y k l m ) , where l m ≪ ε as long as m ≫
1. Later we will see that the perturbation term O ( k Y k l m ) will not damage the qualitative properties of the following system(4.7) H = 12 h Y t , Y i + Z ( X ) + Z ( X ) + εZ ( X , X ) , SYMPTOTIC TRAJECTORIES OF KAM TORUS 37
So we take this (4.7) as our ‘model system’ and research it instead.
Remark . From the following analysis we can see that D h (0) being diagonal isenough. But we still take C3 condition for convenience of our symbolism.Since we can take ε a priori small, so this system(4.8) H = 12 h Y t , Y i + Z ( X ) + Z ( X )will be involved and H can be seen as its O ( ε ) perturbation, ε ≪
1. This ‘weak-coupled’ structure is enlighten by the classical Melnikov method [50], which wasfirstly discovered by V. Melnikov in 1960s. Systems of this kind have several fineproperties from the viewpoint of Mather Theory. We will elaborate these in thefollowing.
Proposition 4.4. • For the model system (4.7), we can find two NHICs N +1 , N +2 corresponding to homology class g = (1 , , g = (0 , . The bottomof N +1 (or N +2 ) is a unique g − (or g − ) homoclinic orbit. Since H is amechanical system, we can also find NHICs N − and N − corresponding to − g and − g , with the bottom − g and − g homoclinic orbits. • Based on our U3 restrictions, we can expand N ± to the minus energysurfaces, i.e. there exists a NHIC N − ∪ N , − e ∪ N +2 , with < e < .Proof. First, we can see that H = Y + Z ( X ) and H = Y + Z ( X ) aretwo uncoupled first-integrals of H . Then at the energy surface S E . = { ( X, Y ) ∈ T ∗ T (cid:12)(cid:12) H = E ≥ } , we can find two 1-dimensional normally hyperbolic tori eachlocated in the foliation S E . = { H = E, H = 0 } and S E . = { H = E, H = 0 } ,which can be denoted by T g ,E and T g ,E . Notice that we can define T − g ,E and T − g ,E in the same way, but we just need to deal with the positive homology caseowing to the symmetry property of mechanical systems. Without loss of generality,we can assume Z + Z take its maximal value 0 at the point (0 , ∈ T . Also wecan transform U3 into a following U3’ for this homogenized case, which is moreconvenient to use:
U3’: − Z ′′ (0) = λ > − Z ′′ (0) = λ >
0. Besides, λ − λ ≥ c > λ λ ≥ c >
1. Here c and c are constants depending only on c and c , andthey are uniformly taken for ∀ m ≥ M ≫ J ∇ H and J ∇ H are independent of each other at theplace S E \ ( T ± g ,E ∪ T ± g ,E ), we can define the stable manifold W sg i ,E and unsta-ble manifold W ug i ,E from the trend of trajectories on it, i = 1 ,
2. Actually theyare invariant Lagrangian graphs and T g i ,E ⊂ W ug i ,E ∩ W sg i ,E . So we can express W s,ug ,E as { ( X, dS s,ug ,E ( X )) + (0 , h ( T g ,E )) (cid:12)(cid:12) X ∈ T × [ − π − δ, π + δ ] ⊂ R } . Here h ( T g ,E ) ∈ R is the average velocity of T g ,E . In the same way we express W s,ug ,E as { ( X, dS s,ug ,E ( X )) + ( h ( T g ,E ) , (cid:12)(cid:12) X ∈ [ − π − δ, π + δ ] × T ⊂ R } . Notice that0 < δ < T in the universal coveringspace R . Actually we can see that S s,ug ,E ( X ) only depends on X and S s,ug ,E ( X )only depends on X in their corresponding domains. So we have † , CHONG-QING CHENG ‡ (1) ∂ ( S ug ,E − S sg ,E )( X ) ∂X (cid:12)(cid:12) T g ,E = 0, ∂ ( S ug ,E − S sg ,E )( X ) ∂X (cid:12)(cid:12) T g ,E = 2 √ λ .(2) ∂ ( S ug ,E − S sg ,E )( X ) ∂X (cid:12)(cid:12) T g ,E = 0, ∂ ( S ug ,E − S sg ,E )( X ) ∂X (cid:12)(cid:12) T g ,E = 2 √ λ .Then the projection set { } × T of T g ,E is the set of minimizers of S ug ,E − S sg ,E .In the same way T × { } is that of ( S ug ,E − S sg ,E )’s. Notice that the former analy-sis is valid for all E ≥
0, and T g i ,E are unique in their small neighborhoods, i = 1 , H now, since ε ≪ a priori small, we can still find the per-turbed manifolds W s,ug i ,E,ε and their generating functions S s,ug i ,E,ε in the correspondingdomains, i = 1 ,
2. Besides, we have S s,ug i ,E,ε = S s,ug i ,E, + εS s,ug i ,E, + O ( ε ) , i = 1 , , and H ( X, ∇ S s,ug i ,E,ε + h ( T εg i ,E )) = E, i = 1 , E > . Here h ( T εg i ,E ) is the average velocity of T εg i ,E . The existence of T εg i ,E ⊂ W sg i ,E,ε ∩ W ug i ,E,ε can be proved by the theorem of implicit function from the two formulasabove. Actually, we can take a section [ −√ ε, √ ε ] ×{ X ∗ } and restrict S ug ,E,ε − S sg ,E,ε on it. We will find the unique minimizer X ( X ∗ ). So we have proved the existenceof T εg ,E with { ( X ( X ∗ ) , X ∗ ) (cid:12)(cid:12) X ∗ ∈ T } its projection. Besides, T εg ,E is hyperbolicand as E changes these tori make up a NHIC corresponding to homology class g .In the same way we get similar results for T εg ,E . Now we raise another restrictionfor Z ( X ). This restriction is just for convenience and is not necessary. C4: Z + Z + εZ reaches its maximum at (0 , ∈ T . Besides, its two eigen-values are still λ and λ , which have the same corresponding eigenvectors with Z + Z at (0 , H = 0 is thesame with H . We denote the energy surface of H by S εE . = { ( X, Y ) ∈ T ∗ T (cid:12)(cid:12) H = E } .Then in the similar way as above we can prove the existence of g − and g − typehomoclinic orbits at S ε , then prove the uniqueness of them.For H system, we can suspend the generating function S g i , into the universalcovering space R . So we have a couple of ˜ S u,s ,~n defined in the domain [ − π − δ, π + δ ] × [ − π − δ, π + δ ] + ~n , where i = 1 , ~n = ( n , n ) ∈ Z . Here + ~ ( · ) is aparallel move in R . Based on C4 , then ˜ S u ,~n − ˜ S s ,~n takes ~n as its unique mini-mizer in the domain of definition. Taking ( X, d ˜ S u ,~n ) as the initial condition, where X ∈ ([ − π − δ, π + δ ] × [ − π − δ, π + δ ] + ~n ), the trajectory of H will exponentiallytend to ( ~n, ∈ T ∗ R as t → −∞ . Similarly, the trajectory with initial condition( X, d ˜ S s ,~n ) tends to ( ~n,
0) exponentially as t → + ∞ .We take { X = π } as the common section and restrict S u , (0 , and S s , (2 π, toit, then we have ∂ ( S u , (0 , − S s , (2 π, )( π, ∂X = 0 , ∂ ( S u , (0 , − S s , (2 π, )( π, ∂X = 2 p λ . SYMPTOTIC TRAJECTORIES OF KAM TORUS 39
Similarly, we take { X = π } as the common section and restrict S u , (0 , and S s , (0 , π ) to it and get ∂ ( S u , (0 , − S s , (0 , π ) )(0 , π ) ∂X = 0 , ∂ ( S u , (0 , − S s , (0 , π ) )(0 , π ) ∂X = 2 p λ . Since H is just a O ( ǫ ) perturbation of H , we also have: S s,u ,~n,ε = S s,u ,~n + εS s,u ,~n, + O ( ε ) , i = 1 , . So ( S u , (0 , ,ε − S s , (2 π, ,ε )( π, X ) has a unique minimizer in [ − π − δ, π + δ ] as asingle variable function of X . Also ( S u , (0 , ,ε − S s , (0 , π ) ,ε )( X , π ) takes its uniquemaximal value in [ − π − δ, π + δ ] as a single variable function of X . Then we get theexistence and uniqueness of g − and g − type homoclinic orbits. The first bulletof this proposition has been proved. Remark . Similar results have been proved by A. Delshams and etc in [18], wherethey call the homoclinic orbits we find ‘isolated’ type and also get the uniquenesswith the same Melnikov method.Second, for the uncoupled system H , we can find a closed trajectory of zero-homology in S − e which is denoted by O − e with a period T − e , e >
0. We can takea section Σ + { X = π,Y > } and restrict it in a small neighborhood of O − e . Here ‘+’means the restriction of Y >
0. Then we have a Poincar´e mapping φ T ( X,Y ) : Σ + { X = π } ∩ B ( O − e , δ ) ∩ S − e → Σ + { X = π } ∩ B ( O − e , δ ) ∩ S − e . Obviously φ T ( X,Y ) has a unique hyperbolic fixed point (0 , π, , p − e − Z ( π )),with the eigenvalues ± √− λ . So we have a expanded NHIC N − ∪ N , − e ∪ N +2 ,where e ∼ O (1) is a proper positive constant.For the system H , we can prove the persistence of N − ∪ N , − e ∪ N +2 as awNHIC via the following theorem 4.7. This wNHIC can be seen as a deformationof N − ∪ N , − e ∪ N +2 . (cid:3) Remark . Notice that the approach of theorem 4.7 is also valid for system H . Theorem 4.7.
There exists ε > sufficiently small such that ∀ < ε ≤ ε , N − ∪ N , − e ∪ N +2 persists as a C wNHIC of system H and satisfies the following: • N − ∪ N , − e ∪ N +2 is C in ε . • N − ∪ N , − e ∪ N +2 ε − close to N − ∪ N , − e ∪ N +2 and can be representedas a graph over it as N − ∪ N , − e ∪ N +2 = { ( X , X , Y , Y ) ∈ T ∗ T (cid:12)(cid:12) X = X ε ( X , Y ) , Y = Y ε ( X , Y ) } with k X ε k C ∼ O ( ε ) and k Y ε k C ∼ O ( ε ) . • There exist locally invariant manifolds W s,uloc ( N − ∪ N , − e ∪ N +2 ) of N − ∪ N , − e ∪ N +2 which are C in ε . † , CHONG-QING CHENG ‡ Figure 11.
Proof.
We can modify H into a new system ˜ H . = H + ερ ( e + He ) Z ( X ), where ρ ( x ) : R → R is a C ∞ function taking value 0 when x ≤ x ≥ H = H restricted in the domain { H ≥ − e } and˜ H = H restricted in { H ≤ − e } . Besides, we have(4.9) k ερ ( 2 e + He ) Z ( X ) k C , {| H |≤ e ∼O (1) } ⋖ O ( ε ) . Since we know that the existence of NHIC N − ∪ N , − e ∪ N +2 , which can be writtenby N , ± ,e for short, we can restrict the tangent bundle on N , ± ,e and split it by T ( T ∗ T ) (cid:12)(cid:12) T N , ± ,e = T N , ± ,e ⊕ T N ⊥ , ± ,e . On the other side, we can calculate the Lyapunov exponents of each of the splittingspaces because H is autonomous and uncoupled. First we will give a definition ofLyapunov exponents for our special case of NHIC.We define the positive Lyapunov exponent restricted on T N , ± ,e by ν + k ( Z ) = lim sup t → + ∞ t ln( k Dφ t ( Z ) · V kk V k ) , and the negative Lyapunov exponent ν −k ( Z ) = lim inf t → + ∞ t ln( k Dφ t ( Z ) · V kk V k ) , where ( Z, V ) ∈ T N , ± ,e and k · k is the Euclid norm. Also we can define the positive (or negative) Lyapunov exponents restricted on T N ⊥ , ± ,e by ν + ⊥ ( Z ) = lim sup t → + ∞ t ln( k Dφ t ( Z ) · V kk V k ) , and ν −⊥ ( Z ) = lim inf t → + ∞ t ln( k Dφ t ( Z ) · V kk V k ) , SYMPTOTIC TRAJECTORIES OF KAM TORUS 41
Figure 12. where (
Z, V ) ∈ T N ⊥ , ± ,e (see [42] for preciser definitions of these). Obviously wecan see that max Z ∈ N , ± ,e ν + k ( Z ) = λ , min Z ∈ N , ± ,e ν −k ( Z ) = − λ , and ν ±⊥ ( Z ) = ± λ , ∀ Z ∈ N ⊥ , ± ,e . There exist ε > ∀ ε < ε , we can finish the proof with thehelp of estimation (4.9) and the following Lemma. (cid:3) Lemma 4.8. (Fenichel, Wiggins) M is a compact, connected C r ( r ≥ em-bedded manifold of R n , which is also invariant of the vector field X . We have thesplitting T M L T M ⊥ = T R n (cid:12)(cid:12) M and ν + ⊥ ν + k > r, ν −⊥ ν −k > r, ∀ x ∈ M. Then ∃ ε > , ∀ Y ∈ B ( X , ε ) vector field, there exists an invariant set M Y which is C r differmorphic to M .Proof. Here we omit the proof since you can find the details in section 7 of [51], orin the paper [23]. (cid:3)
For the system H , we can similarly use Theorem 4.7 and get the persistence ofwNHIC N − ∪ N , − e ∪ N +2 . That’s because ∀ m ≥ M ≫ l − m ≪ ε . Besides, as aperturbation function of ( X, Y ) variables, k ǫR k C ≤ K d ∗ σ/ m l m ( r +6+2 ξ ) / . † , CHONG-QING CHENG ‡ Figure 13.
Corollary 4.9. ∃ M ≫ and ∀ m ≥ M , there exists a wNHIC N − ∪ N , − e ∪ N +2 corresponding to system H , for which the same properties hold as Theorem 4.7.Remark . We actually don’t care the exact value of e , since the locally con-necting orbits are just constructed in energy surfaces with the energy larger thanthe Ma˜n´e Critical Value. Remark . For the case of sub 2-resonant points, it’s enough for us to get thepersistence of this wNHIC N − ∪ N , − e ∪ N +2 . That’s because there isn’t any tran-sition between resonant lines we will adopt a ‘transpierce’ mechanism (see Figure12). But for the case of ω m, / , Lemma 4.8 is invalid for us to prove the persistenceof wNHIC of a g − type. That’s because the Lyapunov exponents ν ±k in this casesatisfies: max Z ∈ N ± k ν ±k ( Z ) k = λ , and k ν ±⊥ ( Z ) k ≡ λ , ∀ Z ∈ N ± . So we need to know more details about the dynamic behaviors of homoclinic orbitsof system H . With these new discoveries we can prove the persistence of N ± wNHICs by sacrificing a small part near the margins. Of course, new restrictionsare necessary and a much preciser calculation will be involved later (see Figure 13). U5: λ λ ∈ R \ Q , ∀ m ∈ N .Based on U5 restriction, we can transform system H into a Normal Form in a B (0 , r ) neighborhood of (0 , ∈ T ∗ T . Here r = r ( λ , λ ) ∼ O (1). SYMPTOTIC TRAJECTORIES OF KAM TORUS 43
Theorem 4.12. (Belitskii [3] , Samovol [46] ) ∀ l ∈ N and ~λ = ( λ , · · · , λ n ) ∈ C n with Re λ i = 0 ( i = 1 , , · · · , n ) , there exists an integer k = k ( l, λ ) such that thefollowing holds: If vector fields V and V have the same fixed point and their jetstill order k coincide with each other, then these two vector fields are C l − conjugate. In our case we have n = 2 and just need l ≥
2. Since our system H is sufficientlysmooth, we can firstly transform it into a normal form of:(4.10) H = λ X Y + λ X Y + O ( X, Y, k + 2) , where ( X, Y ) ∈ B (0 , r ) and k = k ( l, λ ) is the order we needed in Theorem 4.12.Then we can find a C l transformation to convert (4.10) into the linear one:(4.11) H = H ,k ( X Y ) + H ,k ( X Y ) , where ( X, Y ) ∈ B (0 , r ) and H i,k ( · ) is a polynomial of order k , i = 1 ,
2. Notice that H ′ ,k (0) = λ , H ′ ,k (0) = λ . Remark . In fact, U5 can be loosened. Now we give an explanation of this for H . More general case can be found in Sec. 2 of [47]. Definition . We call a vector ~λ = ( λ , λ ) ∈ R k − nonresonant if ∀ ~m =( m , m , m , m ) ∈ N with P i =4 i =1 m i ≤ k , m = m and m = m , we have( m − m ) λ + ( m − m ) λ = 0 . If λ is k − nonresonant with k ( l, λ ) satisfying Theorem 4.12, then we can also finda smooth transformation to convert H into a form of (4.10) in a small neighborhoodof 0. So we actually loosen U5 into the following: U5’: λ is k − nonresonant, ∀ m ∈ N .For system (4.11), we can get the local stable (unstable) manifolds of (0 , ∈ T ∗ T by W sloc = { (0 , , Y , Y ) ∈ B (0 , r ) } , W uloc = { ( X , X , , ∈ B (0 , r ) } . We can further get the parameter function Y = ˆ CY λ λ for trajectories in W sloc and X = ˇ CX λ λ for trajectories in W uloc . On the other side, we can translate (4.11)into a Tonelli form(4.12) H ( Q, P ) = H ,k ( P − Q H ,k ( P − Q , ( Q, P ) ∈ B (0 , r )via (cid:18) X i Y i (cid:19) = √ √ − √ √ ! · (cid:18) Q i P i (cid:19) , i = 1 , . This system is more convenient for us to compare the action value of trajectories.From Proposition (4.4) we know there exists a unique g − type homoclinic orbitas the bottom of NHIC N +1 , which can be denoted by γ . Without loss of general-ity, we can project it onto the configuration space T and then suspend it in theuniversal space R . So in the basic domain (0 , π ) × (0 , π ), γ tends to (0 ,
0) as t → −∞ and tends to (0 , π ) as t → + ∞ (see Figure 14). Besides, we need γ leaves † , CHONG-QING CHENG ‡ Figure 14. (0 ,
0) along the direction ∂ Q and raise a new uniform restriction: U6:
Under the canonical coordinations of (
Q, P ) in the small neighborhood B (0 , r ) of (0 , ∈ R , γ leaves (0 ,
0) according with the trajectory function: Q = ˆ CQ λ λ , < ˆ C ≤ c ( ε ) , which is valid for all m ∈ N .To make U6 satisfied, we just need to restrict Z of a certain form in the domain B ([0 , π ] ×{ } , √ ε ) ∩B ((0 , , r ). Since in B ((0 , , r ) the normal form (4.12) is valid,the coordinate of γ should satisfy Q γ ( t ) = P γ ( t ) , Q γ ( t ) = P γ ( t ) , for t ≤ − t such that ( Q γ ( t ) , Q γ ( t ) , P γ ( t ) , P γ ( t )) ∈ B ((0 , , r ). The extreme caseis that Q γ ≡
0. Once this happens, γ will leave (0 ,
0) along the direction ∂ Q , i.e.ˆ C = Q γ = + ∞ in a rough meaning. Now we will make a local surgery to make ˆ C finite.Recall that ( Q γ ( t ) , P γ ( t )) ∈ W u (0 , ∩ W s (2 π, for all t ∈ R . We can take ε sufficiently small such that W s (2 π, is a graph covering the domain B ([0 , π ] ×{ } , √ ε ) ∩ ( B (0 , r ) \ B (0 , r )). Besides, we know that γ is the unique g − homoclinicorbits from Proposition 4.4. So we just need to change the intersectional point of W u (0 , and W s (2 π, in this domain (see Figure 14). The following Lemma will helpus to achieve this. SYMPTOTIC TRAJECTORIES OF KAM TORUS 45
Figure 15.
Lemma 4.15. (Figalli, Rifford [24] ) Let H : T ∗ M → R be a Tonelli Hamiltonianof class C k with k ≥ . ( Q ( · ) , P ( · )) is a solution of H ( Q, P ) = 0 for t ∈ [0 , T ] , andsatisfies the following: • ( Q (0) , P (0)) = (0 , , Q ( T ) = T and ˙ Q (0) = ∂ Q = ˙ Q ( T ) . • k ˙ Q ( t ) − ∂ Q k ≤ , ∀ t ∈ [0 , T ] .Then ∀ ς > and ε > , ∃ δ = δ ( ς, ε ) such that ∀ Q , P , Q f , P f satisfying Q = (0 , Q , ) , Q f = ( T, Q f, ) , k Q , k ≤ δ, k P − P (0) k ≤ δ, we have k ( T, Q f, ) − ( T, Q , ( T ( Q , P ))) k , k Q f − P ( T ( Q , P )) k < ςε,H ( Q , P ) = H ( Q f , P f ) = 0 , where T ( Q , P ) satisfying Q , ( T ( Q , P )) = T is the arrival time of flow ( Q ( t ) , P ( t )) (see figure 15). There exists a time T f > , a constant K > and a potential V : M → R of class C k such that: • supp ( V ) ⊂ C ( Q , T ( Q , P ) , ς ) ; • k V k C ≤ Kε ; • k T f − T ( Q , P ) k < Krε ; • φ T f H + V ( Q , P ) = ( Q f , P f ) .Here C ( Q , T ( Q , P ) , ς ) is the tube neighborhood defined as C ( Q , T ( Q , P ) , ς ) = { Q ( t ) + (0 , y ) (cid:12)(cid:12) t ∈ [0 , T ( Q , P )] , k y k ≤ ς } , and φ tH + V ( · ) is the Hamiltonian flow of H + V . We can use this Lemma to modify the homoclinic orbits in the open set Ω (thegray set of figure 14). Assuming that γ leaves (0 ,
0) along the direction ∂ Q , then † , CHONG-QING CHENG ‡ we have Q γ ≡
0. Besides, we assume γ ( t ) ∈ Ω ∀ t ∈ [ a, b ]. Then we have T = ( b − a )and two sections,Σ Q γ ( a ) . = { Q = Q γ ( a ) } ∩ Ω , Σ Q γ ( b ) . = { Q = Q γ ( b ) } ∩ Ω , as is showed in figure 15. If ς ∼ O (1) and ε ≪ Q ∈ Σ Q γ ( a ) with k Q − Q γ ( a ) k = δ ( ς, ε ), such that Q ( a + T ( Q , P )) ∈ Σ Q γ ( b ) and k Q f − Q ( a + T ( Q , P )) k < ςε, where Q f = ( Q γ ( b ) ,
0) and ( Q γ ( b ) , , Q γ ( b ) , ∈ W s (2 π, . Then we can take V supported in C ( Q , T ( Q , P ) , ς ) (the gray set of figure 15) and the flow of H + V connects ( Q , P ) and ( Q f , P f ) by Lemma 4.15. Since B ((0 , , r ) ∩ supp ( V ) = ∅ ,the normal form of H + V is still of the form (4.12) and then we have Q , = c ( δ ) Q λ λ , , where c depends only on ε . So we can find a c ε ) constant with c ≤ c and U6 can be satisfied.We can repeat this process in the domain B ([0 , π ] × { } , √ ε ) ∩ ( B ((2 π, , r ) \B ((2 π, , r )) and modify γ to approach (2 π,
0) along the direction ∂ Q . Accord-ingly, we have: U7:
Under the canonical coordinate of (
Q, P ) in the small neighborhood B ((2 π, , r )of (0 , ∈ R , γ approaches (2 π,
0) according to the trajectory function: Q = ˇ CQ λ λ , < ˇ C ≤ c ( ε ) , which is valid for all m ∈ N . Remark . Based on these two restrictions, the unique g − homoclinic orbit willapproach (and leave) the hyperbolic fixed point along the direction of ∂ Q . In thephase space, ( ˙ Q ( ±∞ ) , ˙ P ( ±∞ )) are parallel to the ± λ − eigenvectors (see Figure16). Recall that k λ k < k λ k from U3’ , so the NHIC N +1 will get extra normalhyperbolicity from the hyperbolic fixed point, so does N − from the symmetry ofmechanical systems. Notice that N +1 are foliated by periodic orbits, so we canget such a conclusion: the nearer the periodic orbit approaches the g − homoclinicorbit, the more normal hyperbolicity it will get as it pass by the small neighborhoodof 0 ∈ T ∗ T . On the other side, the nearer the periodic orbit approaches thehomoclinic orbit, the longer its periods is. So we need a precise calculation ofthis competition relationship to persist as large as possible part of NHIC for theultimate system H .As mentioned before, N +1 is foliated by a list of periodic orbits written by γ E with different periods T E . We use the subscript ‘E’ to remind the readers in whichenergy surface γ E lies. Lemma 4.17.
When E ≪ c ( ε ) sufficiently small, we can estimate the period by T E = − C E λ ln Ec , where C E ∼ O (1) is uniformly bounded for all E ∈ [0 , E ] . SYMPTOTIC TRAJECTORIES OF KAM TORUS 47
Figure 16.
Proof.
That’s estimation was gotten in [16] with a tedious but simple computation.Here we just explain the idea of that. When E sufficiently small, the time for γ E passing by B (0 , c ) will be much longer than what it spends outside. Besides,normal form (4.12) is available in this domain B (0 , c ). As long as γ E approachesthe homoclinic orbit γ closely, U6,7 will restrict the enter position ( Q E (0) , P E (0))and exit position ( Q E ( T ) , P E ( T )) of γ E effectively. Here we take c properly smallcomparing to c and assume T as the time of γ E staying in B (0 , c ). On the otherside, we have E = H ,k ( P (0) − Q (0) H ,k ( P (0) − Q (0) E = H ,k ( P ( T ) − Q ( T ) H ,k ( P ( T ) − Q ( T ) . So we can get an estimate of E and T , since the 1-order term of H i,k occupies themain value of the right side of above formulas, i = 1 , T = 1 λ ln c E + τ c , where τ c ∼ O (1) is uniformly bounded for E ∈ [0 , c ]. So we get the periodestimation of a form T E = − C E λ ln Ec and Lemma is proved. (cid:3) Recall that Q γ k Q γ k → (0 ,
1) as t → ±∞ from U6,7 . Then for sufficiently smallenergy E , we can find a couple of 2-dimensional sectionsΣ ± E,ζ . = { ( Q, P ) ∈ R (cid:12)(cid:12) k ( Q, P ) k ≤ c , H ( Q, P ) =
E, Q = ± ζ } , E ∈ [0 , E ] . Then on the energy surface H − (0), W u will intersect Σ +0 ,ζ transversally of a 1-dimensional submanifold, which can be written by Γ u, +0 ,ζ . Similarly W s intersectsΣ − ,ζ of a 1-dimensional submanifold Γ s, − ,ζ . We also know that γ passes across Σ ± ,ζ † , CHONG-QING CHENG ‡ Figure 17. and denote the intersectional points by z ± ,ζ ∈ T ∗ T . It’s obvious that z +0 ,ζ ∈ Γ u, +0 ,ζ and z − ,ζ ∈ Γ s, − ,ζ . So we could define a global mapping(4.14) Ψ ,ζ : Σ +0 ,ζ → Σ − ,ζ with Ψ ,ζ ( z +0 ,ζ ) = z − ,ζ (see Figure 17). From the λ − Lemma [41] we know thatΨ ,ζ (Γ u, +0 ,ζ ) is C − close to Γ u, − ,ζ at the point z − ,ζ , where Γ u, − ,ζ . = W u ∩ Σ − ,ζ . This isbecause Ψ ,ζ (Γ u, +0 ,ζ ) ⊂ Σ − ,ζ and Γ u, +0 ,ζ ⊂ W u , so we have:(4.15) kh D Ψ ,ζ ( z +0 ,ζ ) · v k D Ψ ,ζ ( z +0 ,ζ ) · v k , v ′ ik ≥ k v ′ k , ∀ v ∈ T z +0 ,ζ Γ u, +0 ,ζ , v ′ ∈ T z − ,ζ Γ u, − ,ζ and(4.16) kh D Ψ − ,ζ ( z − ,ζ ) · w k D Ψ − ,ζ ( z − ,ζ ) · w k , w ′ ik ≥ k w ′ k , ∀ w ∈ T z − ,ζ Γ s, − ,ζ , w ′ ∈ T z +0 ,ζ Γ s, +0 ,ζ , provided ζ > λ − Lemmafor 2-dimensional mappings, which can be helpful to reader’s understanding (see
SYMPTOTIC TRAJECTORIES OF KAM TORUS 49
Figure 18.
Figure 18).For sufficiently small
E >
0, Σ ± E,ζ is C r − -close to Σ ± ,ζ . Let z ± E,ζ be intersectionalpoints of γ E with Σ ± E,ζ , we can similarly define(4.17) Ψ
E,ζ : Σ + E,ζ → Σ − E,ζ with Ψ
E,ζ ( z + E,ζ ) = z − E,ζ . Because of the smooth dependence of ODE solutions oninitial data, there exists a sufficiently small ν > ∀ vector v ∈ T z + E,ζ Σ + E,ζ which is ν − parallel to T z +0 ,ζ Γ u, +0 ,ζ in the sense that kh v, v ik ≥ (1 − ν ) k v kk v k holdsfor any v ∈ T z +0 ,ζ Γ u, +0 ,ζ , we have(4.18) kh D Ψ E,ζ ( z + E,ζ ) · v k D Ψ E,ζ ( z + E,ζ ) · v k , v ′ ik ≥ k v ′ k , ∀ v ′ ∈ T z − ,ζ Γ u, − ,ζ . Similarly ∀ w ∈ T z − E,ζ Σ − E,ζ which is ν − parallel to T z − ,ζ Γ s, − ,ζ , we have(4.19) kh D Ψ − E,ζ ( z − E,ζ ) · w k D Ψ − E,ζ ( z − E,ζ ) · w k , v ′ ik ≥ k v ′ k , ∀ v ′ ∈ T z +0 ,ζ Γ s, +0 ,ζ . Once ζ is fixed, we have c ( ζ ) > c − ≤ k D Ψ ,ζ ( z +0 ,ζ ) (cid:12)(cid:12) T z +0 ,ζ Γ u, +0 ,ζ k , k D Ψ − ,ζ ( z − ,ζ ) (cid:12)(cid:12) T z − ,ζ Γ s, − ,ζ k ≤ c . Clearly c → ∞ as ζ →
0. For sufficiently small
E >
0, we have(4.21) (2 c ) − ≤ k D Ψ E,ζ ( z + E,ζ ) v kk v k , k D Ψ − E,ζ ( z − E,ζ ) w kk w k ≤ c , where v is ν − parallel to T z +0 ,ζ Γ u, +0 ,ζ and w is ν − parallel to T z − ,ζ Γ s, − ,ζ .Besides, for E >
E,ζ : Σ − E,ζ → Σ + E,ζ , with Φ E,ζ ( z − e,ζ ) = z + E,ζ . Since ζ ≪ r , normal form (4.12) is available and thetime from z − E,ζ to z + E,ζ is about T = λ ln ζ E + τ ζ (see formula (4.13)), where τ ζ is uniformly bounded as ζ →
0. On the other side, as (4.12) is uncoupled, foran arbitrary vector v ν − parallel to T z − ,ζ Γ u, − ,ζ , D Φ E,ζ ( z − E,ζ ) v is also ν − parallel to † , CHONG-QING CHENG ‡ T z +0 ,ζ Γ u, +0 ,ζ . Analogously, for all w which is ν − parallel to T z +0 ,ζ Γ s, +0 ,ζ , D Φ − E,ζ ( z + E,ζ ) w isalso ν − parallel to T z − ,ζ Γ s, − ,ζ . Furthermore, we have(4.22) c − ( ζ E ) λ λ − µ ≤ k D Φ E,ζ ( z − E,ζ ) v kk v k ≤ c ( ζ E ) λ λ − µ and(4.23) c − ( ζ E ) λ λ − µ ≤ k D Φ − E,ζ ( z + E,ζ ) w kk w k ≤ c ( ζ E ) λ λ − µ , where c > O (1) constant and µ → ζ → E,ζ ◦ Φ E,ζ : Σ − E,ζ → Σ − E,ζ , with Ψ E,ζ ◦ Φ E,ζ ( z − E,ζ ) = z − E,ζ . Then we have(4.25) (2 c c ) − ( ζ E ) λ λ − µ ≤ k D (Ψ E,ζ ◦ Φ E,ζ )( z − E,ζ ) v kk v k ≤ c c ( ζ E ) λ λ − µ for v ν − parallel to T z − ,ζ Γ u, − ,ζ and(4.26) (2 c c ) − ( ζ E ) λ λ − µ ≤ k D (Ψ E,ζ ◦ Φ E,ζ ) − ( z − E,ζ ) w kk w k ≤ c c ( ζ E ) λ λ − µ for w ν − parallel to T z − ,ζ Γ s, − ,ζ .From these two inequalities above, we can see that z − E,ζ is a hyperbolic fixed pointof Ψ ◦ Φ E,ζ . We denote by W sE ( W uE ) the stable (unstable) manifold correspondingto γ E . Besides, we also haveΓ u, ± E,ζ . = W uE ∩ Σ ± E,ζ , Γ s, ± E,ζ . = W sE ∩ Σ ± E,ζ . From
U5,6 , we can take E ≤ (2 c c ) − ι ζ λ ιλ then get the following Lemma 4.18. ∀ E ∈ (0 , E ] , the recurrent mapping Ψ E,ζ ◦ Φ E,ζ satisfying • k D (Ψ E,ζ ◦ Φ E,ζ )( z − E,ζ ) v uE k ≥ E − ( λ λ − ι ) k v uE k , ∀ v uE ∈ T z − E,ζ Γ u, − E,ζ , • k D (Ψ E,ζ ◦ Φ E,ζ ) − ( z − E,ζ ) v sE k ≥ E − ( λ λ − ι ) k v sE k , ∀ v sE ∈ T z − E,ζ Γ s, − E,ζ ,where ι < λ λ and ι ∼ O (1) , as long as ζ is chosen sufficiently small. For E < E , The segment of NHIC N +1 ,E ,E is a 2-dimensional symplectic submanifold. We can restrict symplectic 2-form ω to N +1 ,E ,E , which is equivalentto the area form Ω. Recall that N +1 ,E ,E is an invariant manifold under the flowmapping φ tH , then det | Dφ H ( z ) | ≡ ∀ z ∈ N +1 ,E ,E and t ∈ R . So the eigenvaluesof Dφ tH ( z ) must appear in pairs of the form λ ( z ) and λ ( z ) − . SYMPTOTIC TRAJECTORIES OF KAM TORUS 51
On the other side, the normal form (4.12) is available in the domain B (0 , c ).Then we have E = λ P − Q ) + λ P − Q ) + O ( Q, P, ˙ Q i = ∂H∂P i = λ P i + O ( Q i , P i , , ˙ P i = − ∂H∂Q i = λ Q i + O ( Q i , P i , , where i = 1 ,
2. Then we havemin k ˙ z E k ≥ c ( λ ) √ E, where z E ( t ) = ( Q E ( t ) , P E ( t )) is the trajectory of γ E and c ∼ O (1) is a constantdepending on λ . Besides, ˙ z E ( t ) is just an eigenvector of Dφ tH ( z E ( t )). Therefore, ∃ c > t ∈ R k Dφ tH ( z E ( t )) v k ≥ √ Ec k v k , max t ∈ R k Dφ tH ( z E ( t )) v k ≤ c √ E k v k holds for each vector v tangent to the periodic orbit z E ( t ), ∀ t ∈ R . In fact, aboveformulae give us a control of k Dφ tH k on N +1 ,E ,E . We can make a comparisonbetween | Dφ tH k N +1 ,E ,E and k D Ψ E,ζ ◦ Φ E,ζ k . Lemma 4.19. N +1 ,E ,E is NHIC under φ tH , where t = λ ln E .Proof. As is known to us from (4.13), τ E is uniformly bounded as E →
0, so T E ≤ λ ln E when E is sufficiently small.On the other side, the tangent bundle of T ∗ T over N +1 ,E ,E admits such a Dφ tH (cid:12)(cid:12) t = T E − invariant splitting T z T ∗ T = T z N + ⊥ ⊕ T z N +1 ,E ,E ⊕ T z N −⊥ , z ∈ N +1 ,E ,E . Besides, √ Ec ≤ k Dφ tH ( z ) v kk v k ≤ c √ E , ∀ v ∈ T z N +1 ,E ,E , (4.29) k Dφ tH ( z ) v kk v k ≥ E − ( λ λ − ι ) , ∀ v ∈ T z N −⊥ , (4.30) k Dφ tH ( z ) v kk v k ≤ E ( λ λ − ι ) , ∀ v ∈ T z N + ⊥ , (4.31)holds for t ≥ λ ln E . Since λ λ > ι + 1 > from U3’ and Lemma 4.18. Then wefinished the proof. (cid:3)
For E ∈ [ E , E ] with E ∼ O (1), we can see that N +1 ,E ,E is also NHICunder φ tH with t ≥ λ ln E . Now we explain this. Recall that we can find ahyperbolic fixed point z E,π ∈ Σ E,π of the recurrent mapping P E : Σ E,π → Σ E,π † , CHONG-QING CHENG ‡ from Proposition 4.4, where Σ E,π = { ( X, Y ) ∈ T ∗ T (cid:12)(cid:12) X = π, k X k ≤ √ ε } is alocal section. Besides, we have k DP E ( z E,π ) v kk v k ≥ exp ( λ − c ( ε )) T E , ∀ v ∈ T z E,π ( W uE ∩ Σ E,π ) , (4.32) k DP E ( z E,π ) v kk v k ≤ exp − ( λ − c ( ε )) T E , ∀ v ∈ T z E,π ( W sE ∩ Σ E,π ) , (4.33)where c ( ε ) → ε →
0. On the other side, ∀ v ∈ T z E,π N +1 ,E ,E ,(4.34) √ E c ≤ k Dφ tH ( z ) v kk v k ≤ c √ E , ∀ t ∈ R . If we take t ≥ λ ln E , then T E < t for all E ∈ [ E , E ]. So we have √ E c ≤ k Dφ tH ( z E,π ) v kk v k ≤ c √ E , ∀ v ∈ T z E,π N +1 ,E ,E , (4.35) k Dφ tH ( z E,π ) v kk v k ≥ E − − c λ )1 , ∀ v ∈ T z E,π ( W uE ∩ Σ E,π ) , (4.36) k Dφ tH ( z E,π ) v kk v k ≤ E − c λ )1 , ∀ v ∈ T z E,π ( W sE ∩ Σ E,π ) . (4.37)As E < E , so we get the normal hyperbolicity of N +1 ,E ,E accordingly. Remark . From the analysis of normal hyperbolicity above, we can see that N +1 ,E ,E is NHIC under φ tH with t ≥ λ ln E . Notice that t → + ∞ as E →
0. This leads to a technical flaw and prevent us from proving the persistence of N +1 ,E ,E with E = 0 for system H , using the classical invariant manifold theory[27]. In the following we will give a precise estimation of E > N +1 ,E ,E by sacrificing a small segment near the bottom.Recall that system H is of a form(4.38) H = 12 h Y t , Y i + Z ( X ) + Z ( X ) + εZ ( X , X ) + ǫR ( X, Y, S ) , where ε = L and ǫ = K d ∗ σ/ m l m ( r +6+2 ξ ) / and we can take m ≥ M ≫ ǫ ≪ ε . Once again we use the self-similar structure. The target of thispart is to get the exact value of E . We can assume it by E = ǫ d with d >
0. Laterthe analysis will give d a proper value. Similar as above or [16], we will modify H into(4.39) H ′ = 12 h Y t , Y i + Z ( X ) + Z ( X ) + εZ ( X , X ) + ǫρ ( H − ǫ d ǫ d ) R ( X, Y, S ) , where ρ ( · ) is the same with the one of Theorem 4.7 (see figure 11). We can seethat H ′ = H in the domain { H ≥ ǫ d } and H ′ = H in the domain { H ≤ ǫ d } . Ifwe can prove that there exists a NHIC under φ tH ′ in the domain { H ≥ ǫ d } , then { γ E = ǫd ( S ) , S } must be the bottom of this NHIC. This invariant cylinder verifiesthe existence of wNHIC for H system, as H ′ = H in the domain { H ≥ ǫ d } . Thisis the main idea to prove the persistence of g − wNHIC for H , which was firstlyused in [16] to prove a similar result. SYMPTOTIC TRAJECTORIES OF KAM TORUS 53
Lemma 4.21.
Let the equation ˙ z = F ǫ ( z, t ) be a small perturbation of ˙ z = F ( z, t ) ,with φ tǫ and φ t the flow determined respectively. Then we have: k φ tǫ ( z ) − φ t ( z ) k C ≤ BA (1 − e − At ) e At , where A = max ≤ s ≤ t {k F ( · , s ) k C , k F ǫ ( · , s ) k C } and B = max ≤ s ≤ t k F ǫ ( · , s ) − F ( · , s ) k C . Here k · k C and k · k C only depend on z − variables, and supp z ( F ε − F ) ⊂ R n .Proof. Let ∆ z ( t ) = z ε ( t ) − z ( t ) and∆ ˙ z = ∇ F ǫ ( ν ( t ) z + (1 − ν ( t )) z ǫ )∆ z + ( F ǫ − F )( z ) , where ν ( t ) ∈ [0 , k ∆ ˙ z k ≤ max k∇ F ǫ kk ∆ z k + max k F ǫ − F k . It follows from Gronwall’s inequality that k ∆ z ( t ) k ≤ BA ( e At − . On the other side, the variational equation of Dφ tλ ( λ = ǫ,
0) is the following: ddt Dφ tλ ( z λ ( t ) ) = ∇ F λ ( z λ ( t )) Dφ tλ ( z λ ( t )) , λ = ǫ, . Therefore, for each tangent vector v ∈ T z λ (0) R n one has k Dφ tλ ( z λ (0)) v k ≤ k v k e At . As for D ( φ tǫ − φ t ), we have ddt ( Dφ tǫ ( z ǫ ) − Dφ t ( z )) = ∇ F ǫ ( z ǫ ) Dφ tǫ − ∇ F ( z ) Dφ t = DF ǫ ( z ( t )) Dφ tǫ ( z ǫ ) + D F ǫ ( µz ǫ + (1 − µ ) z )∆ zDφ tǫ − DF ( z ( t )) Dφ t ( z )= DF ǫ ( z )( Dφ tǫ − Dφ t ) + ( DF ǫ ( z ) − DF ( z )) Dφ t + D F ǫ ( µz ǫ + (1 − µ ) z )∆ zDφ tǫ . Then we have k ddt ( Dφ tǫ ( z ǫ ) − Dφ t ( z )) k ≤ A k Dφ tǫ ( z ǫ ) − Dφ t ( z ) k + Be At + B ( e At − e At ≤ A k Dφ tǫ ( z ǫ ) − Dφ t ( z ) k + Be At , and k Dφ tǫ − Dφ t k ≤ BA ( e At − e At )from Gronwall inequality again. Then we complete the proof. (cid:3) † , CHONG-QING CHENG ‡ Now we use this Lemma for our systems H ′ and H . We have the estimation(4.40) k Dφ tH ′ − Dφ tH k C ⋖ ǫ − d ( e c t − e c t ) , where c ∼ O (1) is a constant depending on E . For t ⋖ ln ǫ d − , we can see that k Dφ tH ′ − Dφ tH k C → ǫ →
0. On the other side, Lemma 4.19 gives a timerestriction for the existence of NHIC N +1 ,E ,E . So we need2 λ ln ǫ − d ⋖ ln ǫ d − if we take E = 2 ǫ d . Then d < is sufficient.Therefore, from the invariant manifold theory of [27] we can see that the sep-arated property of spectrums in Lemma 4.19 can not be destroyed as a result of(4.40), as long as ǫ is chosen sufficiently small. So we get the persistence of wNHIC N , ǫd ,E for system H .4.4. Location of Aubry sets.
From subsection 1.3 we can see that there exists a conjugated Lagrangian system L H for any Tonelli Hamiltonian H . The autonomous system H = h + Z has thecorresponding L which satisfies L ( x, v ) . = max p ∈ T ∗ x T {h v, p i − H ( x, p ) } , v ∈ T x T . Besides, it’s a diffeomorphism that L : T T → T ∗ T via ( x, v ) → ( x, p ), where v = H p ( x, p ). Then we can list all the L as follows:(4.41) L ( x, v ) = l ( v, p ∗ i ) − d ∗ σi Z ( x ) , (1-resonance),(4.42) ˜ L ( X, V ) = 12 h V t − (0 , ω i, µι m δ ) , D h − ( p i )( V − (cid:18) ω i, µι m δ (cid:19) ) i − ˜ Z i ( X ) , (homogenized system of transitional segment),(4.43) L ( X, V ) = 12 V − Z ( X ) − Z ( X ) − εZ ( X ) , (homogenized system of 2-resonance),(4.44) L ( X, V ) = 12 V − Z ( X ) − Z ( X ) , (uncoupled system of 2-resonance).Recall that these autonomous Lagrangians are all Tonelli and defined in the domainswhere canonical coordinations are valid respectively. For a given ~h = (0 , h ) ∈ H ( T , R ), the minimizing measure µ h exists and supp( µ h ) lays on the cylinder N ( N for L ). Actually, µ h is uniquely ergodic with the periodic trajectory γ h as itssupport. This is because the strictly positive definiteness of h ( p ) w.r.t. p .Besides, we have β L ( h ) = Z Ldµ h . SYMPTOTIC TRAJECTORIES OF KAM TORUS 55
Since α L ( c ) is the convex conjugation of β L ( h ), we can find c h = (0 , c h, ) ∈ D + β L ( h ) ∈ H ( T , R ) and α L ( c h ) = − Z L − c h dµ h = H (cid:12)(cid:12) supp ( µ h ) , i.e. µ h is the c h − minimizing measure.Besides, we can see that ˜ A L ( c h ) = f M L ( c h ) =supp( µ h ). Then e N L ( c h ) = ˜ A L ( c h )from [4]. Since ˜ N L ( c h ) is upper semicontinuous as a set-valued function of L (seesubsection 1.3), then ∀ ǫ > δ ( ǫ ) such that for k L − L k C ≤ δ , e N L ( c h ) ⊂ B ( e N L ( c h ) , ǫ ). However, in the previous paragraph wehave proved the persistence of wNHIC for L conjugated to H = H + R . As thewNHIC is the unique invariant set in its small neighborhood (the normal hyperbolicproperty), this implies that ˜ A L ( c h ) is still contained in the weak invariant cylinder N .For the 1-resonant case, we can see that(4.45) L = L + R ( x, v, t ) , where k R ( x, v, t ) k C , B ⋖ d σ − r − m . Owing to our self-similar structure, we can take m ≥ M ≫ e N L ( c h ) is in the wNHIC N . Actually, thisis a typical a priori unstable case. the readers can find an alternative proof from[8] and [16].For the transitional segment from 1-resonance to 2-resonance, we can see that(4.46) L = ˜ L + R ( X, V, S ) , where k R k C , B ⋖ K . Recall that ˜ H is homogenized and the norm k · k C , B dependsjust on ( X, V ) variables, but not on S . However, we needn’t control the value of ∂ S R in the Legendre transformation from H to L . So we can take K a priori largesuch that e N L ( c h ) is in the wNHIC N due to above analysis. Theorem 4.22.
Given g = (0 , ∈ H ( T , Z ) , there exists a wNHIC N ⊂ T ∗ T × S corresponding to it for the two cases above: 1-resonance and transi-tional segment. We can find a wedge region W cg ⊂ H ( T , R ) corresponding to [ h , h ] ⊂ H ( T , R ) with h i = (0 , h i, ) = ν i g , i = 1 , . ∀ c h = (0 , c h, ) ∈ W g for h ∈ [ h , h ] , we have ˜ A L ( c h ) ⊂ e N L ( c h ) ⊂ N . Besides, from the upper semicontin-uous of Ma˜n´e set, we can see that e N L ( c ′ h ) ⊂ N for c ′ h = ( c ′ h, , c h, ) ∈ ˚ W g Now we consider the 2-resonant case of ω m, / . We also need to consider the lo-cation of Aubry sets for N cylinder in this case, as there exists a transformation ofresonant lines. First, the uncoupled Lagrangian L has two routes Υ hg . = { (0 , h ) ∈ H ( T , R ) (cid:12)(cid:12) h ∈ [0 , h max2 ] } and Υ hg . = { ( h , ∈ H ( T , R ) (cid:12)(cid:12) h ∈ [0 , h max1 ] } , of whichwe can find minimizing measures µ (0 ,h ) and µ ( h , respectively. Besides, theyare uniquely ergodic with the supports lay on cylinders N and N . Accord-ingly, we can find routes Υ cg . = { (0 , c ) ∈ H ( T , R ) (cid:12)(cid:12) c ∈ [0 , c max2 ] } and Υ cg . = { ( c , ∈ H ( T , R ) (cid:12)(cid:12) c ∈ [0 , c max1 ] } , of which µ (0 ,h ) is (0 , c ) − minimizing and † , CHONG-QING CHENG ‡ Figure 19. µ ( h , is ( c , − minimizing. Similarly as above, e N L ( c ) = e A L ( c ) = f M L ( c ) ⊂ N i for c ∈ Υ cg i , i = 1 , L can be considered as an autonomous perturbation of L by taking ε ≪ a priori small. From C4 we can see that min c α L ( c ) = 0 = α L (0). Still bythe upper semicontinuous of Ma˜n´e set, e N L ( c ) is contained in a small neighborhoodof e N L ( c ), of which the neighborhood radius depends only on ε . Then we have e N L ( c ) ⊂ N i for c ∈ Υ cg i , i = 1 , e N L (0) = e A L (0) = f M L (0) = { } ⊂ T ∗ T . So we have a 2-dimensionalflat F containing 0 ∈ H ( T , R ) in it and e N ( c ) = { } , ∀ c ∈ ˚ F . So we can find(0 , c min2 ) and ( c min1 ,
0) in ∂ F such that e N ((0 , c min2 )) = { } ∪ { γ g } , and e N (( c min1 , { } ∪ { γ g } . These are the end points of routes Υ cg and Υ cg of which Ma˜n´e set is contained inthe cylinders N or N respectively (see figure 19). As a further perturbation of L , we have:(4.47) L = L − R ( X, V, S ) , where k R k C , B ∼ O ( ǫ ). Since we have proved that N can be expanded to minusenergy surfaces from Corollary 4.9, we can still find a route Υ cg = { (0 , c ) (cid:12)(cid:12) c ∈ [ c ∗ min2 , c ∗ max2 ] } of which the Ma˜n´e set is contained in N . Besides, there is still aflat F ∗ containing 0 ∈ H ( T , R ) in it and (0 , c ∗ min2 ) ∈ ∂ F ∗ .On the other side, we only proved the persistence of wNHIC N ,E ,E with E = 2 ǫ d in the previous paragraph. So we can find a lower bound c ∗∗ min1 and SYMPTOTIC TRAJECTORIES OF KAM TORUS 57
Figure 20. Υ cg = { ( c , (cid:12)(cid:12) c ∈ [ c ∗∗ min1 , c ∗ max1 ] } , of which e N ( c ) ⊂ N ,E ,E . Notice that( c ∗∗ min1 ,
0) doesn’t lie on F ∗ (see Figure 20).We can find a c min1 corresponding to c ∗∗ min1 and α L (( c min1 , ǫ d . Withoutloss of generality, we assume µ ( c min1 , is the corresponding minimizing measure of L . Then we have α L ( c min1 , ≥ ǫ d − O ( ǫ ) ≫ ǫ d ≥ E . So we can see that theminimizing measure µ ( c min1 , lays on the wNHIC N ,E ,E . In contrary, this sup-plies a upper bound of c min1 > c ∗∗ min1 .Similarly as above, from the upper semicontinuous of e N L ( c ) as a set-valuedfunction of c , there exist two wedge set W cg and W cg satisfying W cg . = { ( c , c w ) (cid:12)(cid:12) c ∈ [ c ∗∗ min1 , c ∗ max1 ] , k c w k ≤ δ ( c ) } and W cg . = { ( c w , c ) (cid:12)(cid:12) c ∈ [0 , c ∗ max2 ] , k c w k ≤ δ ( c ) } . Here δ > c and c can be properly chosen, but we actually don’tcare the exact value of δ . Theorem 4.23.
For the case of 2-resonance, we can find 3-dimensional cylinder N i corresponding to g i ∈ H ( T , Z ) , i = 1 , . There exists a wedge set W cg i as isgiven above such that • ∀ c ∈ ˚ W g i , ˜ A ( c ) ⊂ e N ( c ) ⊂ N i . • the wedge set reaches to certain small neighborhood of flat F ∗ in the sensethat min c ∈ W cg α ( c ) − min c ∈ H ( T , R ) α ( c ) ≤ ǫ d , † , CHONG-QING CHENG ‡ and min c ∈ W cg α ( c ) − min c ∈ H ( T , R ) α ( c ) = 0 . Therefore, we can connect different W cg sets into a long ‘channel’ according toΓ ωm, . Recall that here a homogenization method is involved in the cases of 2-resonance and transition part from 1-resonance to 2-resonance. But this approachis just a special kind symplectic transformation. The following subsection ensuresthe validity of all these properties for system H in the original coordinations.4.5. Symplectic invariance of Aubry sets.
Let ω = dp ∧ dq be the symplectic 2-form of T ∗ M . The diffeomorphismΨ : T ∗ M → T ∗ M, via ( q, p ) → ( Q, P )is call exact , if Ψ ∗ ω − ω is exact 2-form of T ∗ M . Theorem 4.24. (Bernard [5] ) For the exact symplectic diffeomorphism Ψ and H : T ∗ M → R Tonelli Hamiltonian, α H ( c ) = α Ψ ∗ H (Ψ ∗ c ) , f M H ( c ) = Ψ( f M Ψ ∗ H (Ψ ∗ c )) , e A H ( c ) = Ψ( e A Ψ ∗ H (Ψ ∗ c )) , e N H ( c ) = Ψ( e N Ψ ∗ H (Ψ ∗ c )) , where ∀ c ∈ H ( M, R ) . Since the time-1 mapping of Hamiltonian flow φ tH (cid:12)(cid:12) t =1 can be isotopic to identityand φ ∗ tH (cid:12)(cid:12) t =1 ω = ω , then φ tH (cid:12)(cid:12) t =1 is exact symplectic. So we can use this theoremand get the symplectic invariance of these sets via different KAM iterations andhomogenization. 5. Annulus of incomplete intersection
From the previous section we can see that W cg can reach the place ǫ d − approachingto arg min c α L ( c ). But it’s unclear whether it can reach the margin of F ∗ . So wehave to find a route in H ( T , R ) to connect W cg and W cg , along which the diffu-sion orbits can be constructed connecting the two wNHICs N and N . In contrary,this demands that there must be an annulus region A around F ∗ of the thicknessgreater than ǫ d such that ∀ c, c ′ ∈ A are c − equivalent (see Figure 25). This is thecentral topic we’ll discuss in this section.We mention that the ‘incomplete intersection’ here means that the stable man-ifold of the Aubry set ‘intersects’ the unstable manifold non-trivially but possiblyincomplete. In other words, for each class in this region, the Ma˜n´e set does notcover the whole configuration space. On the other side, [32] supplies us with amechanism to construct c − equivalent in this region, which we will discuss in thenext section. SYMPTOTIC TRAJECTORIES OF KAM TORUS 59 α − flat of system H . For H system, we have α H (0) = 0 and there exists a 2-dimensional flat F suchthat 0 ∈ ˚ F . Besides, we have the following Proposition 5.1. (1) ∀ c ∈ ˚ F , e N ( c ) = ˜ A ( c ) = f M ( c ) = { (0 , ∈ T ∗ T } . (2) There exist two sub-flat E i ⊂ ∂ F , such that ∀ c ∈ E i f M ( c ) = { (0 , } , e N ( c ) = ˜ A ( c ) = { (0 , } ∪ { γ g i } , i = 1 , . (3) ∀ c ∈ ∂ F , { (0 , } ⊂ f M ( c ) and e N ( c ) \ { (0 , } 6 = ∅ . (4) ∀ c ∈ ∂ F , if we can find more than one ergodic minimizing measure µ c and µ ′ c , there exists γ c ⊂ e N ( c ) connecting supp µ c and supp µ ′ c .Proof.
1. A direct cite of [35].2. We can see this point from the analysis of previous subsection.3. See [52] for the proof.4. A direct cite of [17]. (cid:3)
In the universal covering space R , we consider the elementary weak KAM so-lution u ± (0 , corresponding to 0 ∈ R , which is the projection of hyperbolic fixedpoint (0 , ∈ T ∗ T . Since ε ≪ W s,u (0 , cancover a whole basic domain of R , i.e. W s,u (0 , = { ( x, dS s,u (0 , ( x )) (cid:12)(cid:12) x ∈ Ω } . Here Ωis the maximal domain of which W s,u (0 , is a graph. We can see that there exists asmall constant δ > ε and [ − π − δ, π + δ ] × [ − π − δ, π + δ ] ⊂ Ω. Lemma 5.2. ∀ x ∈ Ω , ( x, du ± (0 , ) = ( x, dS s,u (0 , ) , i.e. u ± (0 , is differentiable in Ω .Proof. We just prove the Lemma for u − (0 , and the case u +(0 , can be proved in thesame way. Because u − (0 , is linear semi-concave (SCL) from [11], so it’s differen-tiable for a.e. x ∈ Ω. If x ∈ Ω is a differentiable point, ( x, du − (0 , ( x )) will decidea backward semistatic trajectory γ − (0 , ( t ) which trends to (0 ,
0) as t → −∞ . Then( γ − (0 , ( t ) , ˙ γ − (0 , ( t )) ⊂ W u (0 , for t ∈ ( −∞ , γ − (0 , (0) = x . Notice that theremay be some t ∈ ( −∞ ,
0] existing and y = γ − (0 , ( t ) ∈ ∂ Ω (see Figure 21). Butthere exists no focus locus for t ∈ ( −∞ ,
0) in the set { ( γ − (0 , ( t ) , ˙ γ − (0 , ( t )) } (see The-orem 6.3.6 of [11]). So u − (0 , is C r differentiable at the set { ( γ − (0 , ( t ) , ˙ γ − (0 , ( t )) (cid:12)(cid:12) t ∈ ( −∞ , } . But on the other side, we can see that D u − (0 , doesn’t exist becausethe invalidity of graph property. This contradiction means that y doesn’t exist and { ( γ − (0 , ( t ) , ˙ γ − (0 , ( t )) } lays on the graph part of W u (0 , for t ∈ ( −∞ , x of which u − (0 , is not differentiable, from [11], for arbitraryreachable gradient p ∈ D ∗ u − (0 , ( x ) we can find a sequence of differentiable points x n → x and ( x n , du − (0 , ( x n )) → ( x, p ), where { x n } ⊂ Ω for n ∈ N . Then ( x, p ) lieson the graph part of W u (0 , . But from the invariant manifold theorem W u (0 , is C r − differentiable, so D ∗ u − (0 , ( x ) is a single-point set and u − (0 , is differentiable atthis point. (cid:3) Based on this Lemma, we can prove the following † , CHONG-QING CHENG ‡ Figure 21.
Theorem 5.3. ∀ c ∈ ∂ F , f M ( c ) = { (0 , ∈ T ∗ T } , i.e. there doesn’t exist newminimizing measures except the one supported on the hyperbolic fixed point.Proof. If ∃ c ∈ ∂ F such that f M ( c ) \{ (0 , } 6 = ∅ , there must be an ergodic measure µ c with ρ ( µ c ) = 0 and supp( µ c ) ∩{ (0 , } 6 = ∅ . From (4) of Proposition 5.1, we can finda c-semistatic orbit γ c connecting supp( µ c ) and { (0 , } . Then ∀ t ∈ ( −∞ , + ∞ ), γ c : ( −∞ , t ] is of course a backward semistatic orbits. Then { ( γ c ( t ) , ˙ γ c ( t )) } lieson the graph part of W u (0 , for all t ∈ ( −∞ , t ] from the above Lemma. On theother side, the hyperbolic fixed point is the only invariant set of W u (0 , (cid:12)(cid:12) Ω and t isarbitrarily chosen, so µ c only can be the fixed point itself. (cid:3) Based on this Theorem, we can use the Theorem 3.2 of [16] and see that F isactually a polygon with finite edges. Besides, F is of central symmetry as H is amechanical system. Then from (3) of Proposition 5.1 we can see that e N ( c ) = ˜ A ( c )and there must be homoclinic orbits come out for c ∈ ∂ F . Without loss of gener-ality, we can assume the edges of F as E ± i and ∀ c ∈ E ± i , there will be homoclinicorbits coming out with homologgy class just being ± g i , i = 1 , · · · , m .Actually, we can prove in the following that the only possible homology classes ofhomoclinic orbits are (0 , , ,
1) and ( − , , − and (1 , − homoclinic orbit has been proved in the previous subsection, we onlyneed to prove the existence of (1 , − type and ( − , − type can be sealed with inthe same way.Suppose ~n = ( n , n ) and γ n is a n − type homoclinic orbit. Accordingly, we have E n as an edge of F and γ n ⊂ N ( c ), ∀ c ∈ E n . Then { ( γ n ( t ) , ˙ γ n ( t )) } is contained in thegraph part of W u (0 , , for t ∈ ( −∞ , t ]. By taking ε ≪ ∩B ((2 π, π ) , δ ), Ω ∩B ((2 π, , δ ) and Ω ∩B ((0 , π ) , δ )are all nonempty. Here Ω is the definition domain of W u (0 , and δ = δ ( λ ) is theradius of the neighborhood of the fixed point, in which the normal form (4.12)is valid (see figure 22). There exists [ t , t ] during which γ n ( t ) is contained in B ((2 π, π ) , δ ). Since (4.12) is uncoupled, the corresponding equation (4.27) is C conjugated to the linear ODE(5.1) ˙ X i = λ i Y i , ˙ Y i = λ i X i , i = 1 , , SYMPTOTIC TRAJECTORIES OF KAM TORUS 61 in the domain B ((2 π, π ) , δ ) from the Hartman Theorem of [26]. Under this coor-dination we can see that H ( X, Y ) = λ ( Y − X )2 + λ ( Y − X )2 , and L ( X, V ) = ( V + λ X )2 λ + ( V + λ X )2 λ , where V i = λ i Y i , i = 1 ,
2. This smoothness is enough for us to calculate theaction value of γ n . As the coordinate transformation doesn’t change the energy, { γ n ( t ) (cid:12)(cid:12) t ∈ [ t ,t ] } lies on the energy surface { H = 0 } . Besides, H i = λ i ( Y i − X i )2 is afirst integral in this domain, i = 1 ,
2. So we can involve a parameter e to simplifyour calculation, where H (cid:12)(cid:12) ( γ n ( t ) , ˙ γ n ( t )) = e and H (cid:12)(cid:12) ( γ n ( t ) , ˙ γ n ( t )) = − e , t ∈ [ t , t ].Without loss of generality, we can assume that γ n ( t i ) = X i , where X i =( X i , X i ) ∈ ∂ B ((2 π, π ) , δ ) with i = 1 ,
2. By a tedious but simple computation,we can get the flow via(5.2) (cid:26) X = a e λ t + a e − λ t ,Y = a e λ t − a e − λ t , (5.3) (cid:26) X = b e λ t + b e − λ t ,Y = b e λ t − b e − λ t , where a = X − e − λ T X e λ T − e − λ T , a = e λ T X − X e λ T − e − λ T , and b = X − e − λ T X e λ T − e − λ T , b = e λ T X − X e λ T − e − λ T , with T = t − t . We then get the action as(5.4) A L ( γ n ) (cid:12)(cid:12) [ t ,t ] = X
11 2 + X
12 2 λ T cosh 2 λ T − X
21 2 + X
22 2 λ T cosh 2 λ T − . Now we connect X i and (2 π, π ) with trajectories lie on W u,s (2 π, π ) . Also we cancalculate the formula as(5.5) I : (cid:26) X i ( t ) = X i e − λ i t ,Y i ( t ) = − X i e − λ i t , where t ∈ [0 , + ∞ ) and i = 1 , II : (cid:26) X i ( t ) = X i e λ i t ,Y i ( t ) = X i e λ i t , where t ∈ ( −∞ ,
0] and i = 1 ,
2. We then get the action(5.7) A L ( I ) + A L ( II ) = X
11 2 + X
12 2 X
21 2 + X
22 2 . Comparing the actions and we get A L ( I ) + A L ( II ) < A L ( γ n ) (cid:12)(cid:12) [ t ,t ] . Therefore γ n will break into two segment when it passes B ((2 π, π ) , δ ), as it’s semi-static.Then γ n can be decomposed into γ (1 , and γ ( n − ,n − . In the same way we can † , CHONG-QING CHENG ‡ Figure 22. decompose γ ( n − ,n − . So we get that all the minimizing homoclinic orbits arejust of these homology classes:(1 , , (1 , , (0 , , ( − , , ( − , , ( − , − , (0 , − , (1 , − . Theorem 5.4. ∂ F only could be rectangle, hexagon and octagon (see figure 23).Proof.
1. As a convex set, the number of expose points of F is not more than thenumber of homology classes of minimizing homoclinic orbits.2. The inner of sub-flat E will share the same homoclinic orbits γ g , and h g, c − c ′ i = 0, ∀ c, c ′ ∈ ˚ E . SYMPTOTIC TRAJECTORIES OF KAM TORUS 63
Figure 23.Figure 24.
3. The homology class of minimizing homoclinic orbit is irreducible, i.e. l · ( n , n ) − type minimizing homoclinic orbit is a conjunction of ( n , n ) − type mini-mizing homoclinic orbits. (cid:3) Since we have proved the uniqueness of ( ± , − and (0 , ± − type minimizinghomoclinic orbits, we just need to reduce the number of minimizing homoclinic † , CHONG-QING CHENG ‡ orbits of other classes. The Melnikov method can be used and we raise a new re-striction U7:
The Melnikov function of H has a unique critical point in B (( π, π ) , δ ). Here δ = δ ( ǫ ) > H can be of (1 , − type and ( − , − type,which is written by M H, (1 , and M H, ( − , . Once U7 is satisfied, then (1 , − typeand ( − , − type homoclinic orbits is unique in B (( π, π ) , δ ) (see figure 24). Since H is a mechanical system, the homoclinic orbit of other type is also unique. Actually,we can satisfy U7 by restricting Z of a certain form in B (( π, π ) , δ ), since theMelnikov function can be considered as a continuous linear functional of potential Z ( X ): M H, (1 , ( X ) = lim T → + ∞ − Z T − T Z ( γ (1 , ( t )) dt, (5.8) M H, ( − , ( X ) = lim T → + ∞ − Z T − T Z ( γ ( − , ( t )) dt, (5.9)where γ (1 , (0) = X = γ ( − , is the homoclinic orbit of H of certain homologyclass, X ∈ B (( π, π ) , δ ). Proposition 5.5. • ∇ H M H, (1 , ( X ) = 0 , ∇ H M H, ( − , ( X ) = 0 . • If for some X ∈ B (( π, π ) , δ ) , we have ∇ H M H, (1 , ( X ) = 0 = ∇ H M H, (1 , ( X ) , and the rank of the matrix ( ∇ H i ∇ H j M H, (1 , ( X )) i,j =1 , equals one. Thenfor sufficiently small ε , there exists a transversal homoclinic orbit of H passing from the O ( ε ) neighborhood of X . For M H, ( − , ( X ) we have thesame conclusion.Proof. See Appendix for the proof. It’s a direct cite of [50]. (cid:3)
Once U7 is satisfied, we have Theorem 5.6. ∀ c ∈ ∂ F , we have N ( c ) $ T . Thickness of Annulus.
From the previous Theorem and the upper semicontinuity of Ma˜n´e set, thereexists ∆ > ∀ c ∈ { α H ( c ≤ ∆ ) } , we have e N ( c ) $ T . Here ∆ is aconstant depending on ε .On the other side, for ∆ ∈ (0 , ∆ ) and c ∈ α − H (∆), All the c − minimizing mea-sures have the same rotational direction because of the graph property of Matherset [12]. So we can find a loop section Γ c ⊂ T such that all the semi-static orbitsintersect it transversally. As is known to us that N H ( c ) $ T , we can find finitelymany open sets { U i } ni =1 covering Γ c ∩ N H ( c ). { U i } ni =1 is diffeomorphic to a list of SYMPTOTIC TRAJECTORIES OF KAM TORUS 65
Figure 25. open internals { ( a i , b i ) ⊂ [0 , } ni =1 and they are disjoint with each other.Once again we use the upper-semicontinuity of Ma˜n´e set, for m ≥ M ≫ H satisfies the following Theorem 5.7. ∃ ǫ > and ∆ > depending on ε , such that ∀ ∆ < ∆ , ǫ ∈ (0 , ǫ ) and c ∈ α − (∆) , N H ( c ) intersects Γ c × { S ≡ } transversally. Besides, we stillhave N H ( c ) ∩ Γ c ×{ S ≡ } ⊂ S ni =1 U i . Here { U i } ⊂ Γ c ×{ S ≡ } are 1-dimensionalopen internals disjoint with each other. On the other side, we can see that the wedge set W cg can reach the place of∆ = 3 ǫ d from Theorem 4.23 (see Figure 25). So the following property of H isvalid: Corollary 5.8. (Overlap Property)
For g and g , ∃ ǫ > sufficiently smallsuch that W cg and W cg could intersect the annulus region A . = { c (cid:12)(cid:12) α H ( c ) ∈ [0 , ∆ ] } ,as long as ǫ ≤ ǫ .Remark . (Robustness of system H ) Recall that our system is actually of aform (4.6), which can be considered as a O ( l m ) perturbation of H . But (4.6) isalso an autonomous system, so all these uniform properties for H can be preservedfor it, as long as we take m ≥ M ≫ a posterior large.6. Local connecting orbits and generic diffusion mechanism
To construct orbits connecting some Aubry set to another Aubry set nearby,we introduce two types of modified Tonelli Lagrangian, i.e. the time-step and thespace-step Lagrangian. The former one was firstly developed in [4], [14] and [15], † , CHONG-QING CHENG ‡ with the earliest idea given by J. Mather in [38]. The latter one was firstly de-veloped in [32]. Then C-Q. Cheng made a further elaboration and generalizationof it in [16], for dealing with a priori stable Arnold Diffusion problem. Since ourconstruction in this section have a great similarity with this case of [16], we chooseit as the main reference for this section.Actually, we can ascribe the local connecting orbits as two different mechanism:Arnold’s and Mather’s. The former one is essentially heteroclinic orbit, which isknown as h − equivalent orbit from the variational viewpoint. The latter one isknown as c − equivalent orbit, which is constructed with the topological conditionsprovided by Aubry sets. We mainly use the Arnold mechanism to construct thediffusion orbits along the wNHICs. The Mather mechanism is mainly used to solvethe difficulty of incomplete intersection annulus.6.1. Modified Lagrangian: time-step case.Definition 6.1. (Time-step Lagrangian)
We call a Tonelli Lagrangian L : T T n × R → R time-step Lagrangian , if we can find L − and L + : T T n × S → R such that L ( · , t ) = L − ( · , t ) , ∀ t ∈ ( −∞ , L ( · , t ) = L + ( · , t ) , ∀ t ∈ [1 , + ∞ ) , i.e. L ( · , t ) is periodic in ( −∞ , ∪ [1 , + ∞ ). Definition 6.2.
A curve γ : R → M is called minimal if Z τ ′ τ L ( γ ( t ) , ˙ γ ( t ) , t ) dt ≤ Z τ ′ τ L ( ζ ( t ) , ˙ ζ ( t ) , t ) dt holds for τ < τ ′ and every absolutely continuous curve ζ : [ τ, τ ′ ] → M with ζ ( τ ) = γ ( τ ) and ζ ( τ ′ ) = γ ( τ ′ ). We denote the set of minimal curves for time-step Lagrangian L by G ( L ) and ˜ G ( L ) . = { ( γ ( t ) , ˙ γ ( t ) , t ) ∈ T M × R (cid:12)(cid:12) γ ∈ G ( L ) } . Thenwe have G ( L ) = π ˜ G ( L ) where π : T M × R → M × R is the standard projection. Theorem 6.3. [14, 15, 16]
The set-valued map L → ˜ G ( L ) is upper semicontinuous.Consequently, the map L → G ( L ) is also upper semicontinuous.Proof. Here we only give a sketch of the proof and more details can be found in[16]. Let { L i } ⊂ C r ( T M × R , R ) be a sequence converging to L under the norm k · k C ,T M , and γ i ∈ G ( L i ) be a sequence of minimal curves. These two modifiedLagrangian are actually defined in a proper universal covering space of T n +1 . Wecan see that k ˙ γ i ( t ) k ≤ K , ∀ t ∈ R with K is a constant depending on k ∂ vv L i k . Thenthe set { γ i } is compact in the C ( R , M ) − topology. Let γ be one accumulated pointof γ i , we can see that γ is L − minimal, and this proves the upper-semicontinuity of˜ G ( L ). (cid:3) In application, the set G ( L ) seems too big for the construction of connectingorbits. For time-step Lagrangian, we can introduce the following set of pseudo SYMPTOTIC TRAJECTORIES OF KAM TORUS 67 connecting orbits, which is written by C ( L ). Let α ± be the minimal average actionof L ± . For m , m ∈ M and T , T >
0, we define h T ,T L ( m , m ) = inf γ ( − T )= m γ ( T )= m Z T − T L ( dγ ( t ) , t ) dt + T α − + T α + and h ∞ L ( m , m ) = lim inf T ,T →∞ h T ,T L ( m , m ) . Let { T i } i ∈ N and { T i } i ∈ N be the sequence of positive numbers such that T ij → ∞ ( j = 0 ,
1) as i → ∞ and satisfieslim i →∞ h T i ,T i L ( m , m ) = h ∞ L ( m , m ) . Accordingly, we can find γ i ( t, m , m ) : [ − T i , T i ] → M being the minimizer con-necting m and m and h T i ,T i L ( m , m ) = Z T i − T i L ( dγ i ( t ) , t ) dt + T i α − + T i α + . Then the following Lemma holds:
Lemma 6.4.
The set { γ i } is pre-compact in C ( R , M ) . Let γ : R → M be anaccumulation point of { γ i } , then ∀ s, τ ≥ A L ( γ ) (cid:12)(cid:12) [ − s,τ ] = inf s − s ∈ Z ,τ − τ ∈ Z s ,τ ≥ γ ∗ ( − s )= γ ( − s ) γ ∗ ( τ )= γ ( τ ) Z τ − s L ( dγ ∗ ( t ) , t ) dt + ( s − s ) α − + ( τ − τ ) α + . (6.1) Proof.
The pre-compactness of { γ i } can be proved in the same way with that of G ( L ). As for the formula 6.1, we can use the proof by contradiction, with anapproach of comparing the action. We omit the proof since you can find it in[16]. (cid:3) We define C ( L ) by the set { γ ∈ G ( L ) (cid:12)(cid:12) (6 .
1) holds for γ } . Clearly, ∀ γ ∈ C ( L ),the orbits ( γ ( t ) , ˙ γ ( t ) , t ) approaches to ˜ A ( L − ) as t → −∞ and approaches to ˜ A ( L + )as t → + ∞ . That’s why we call it pseudo connecting curve. We also have e C ( L ) = [ γ ∈ C ( L ) ( γ ( t ) , ˙ γ ( t ) , t ) , C ( L ) = [ γ ∈ C ( L ) ( γ ( t ) , t ) . If L is a periodic Tonelli Lagrangian, then ˜ C ( L ) = e N ( L ) and C ( L ) = N ( L ). Theorem 6.5. [14, 15, 16]
The map L → C ( L ) is upper semicontinuous. As thespecial case for periodic Lagrangian L , c → N ( L ) as well as the map c → e N ( L ) isupper semicontinuous. Corollary 6.6.
For periodic Tonelli Lagrangian L and c ∈ H ( M, R ) , the set-valued function c → N ( c ) is upper semicontinuous. † , CHONG-QING CHENG ‡ Modified Lagrangian: space-step case.Definition 6.7. (Space-step Lagrangian)
In the covering space
M . = R × T n − of T n , we call a Tonelli Lagrangian L : T M × S → R space-step , if we can find L − and L + : T T n × S → R such that L ( x , · ) = L − ( x , · ) , ∀ x ∈ ( −∞ , L ( x , · ) = L + ( x , · ) , ∀ x ∈ [1 , + ∞ ) , where ( x , x , · · · , x n , t ) ∈ M × S . Besides, the following conditions should besatisfied at the same time: • Let µ ± be the 0 − minimizing measure of L ± respectively. Then π ρ ( µ ± ) > µ ± . • α − = α + as the minimal average action of L ± . Without loss of generality,we assume it equals 0. • k L + − L − k { ( x,v ) ∈ T M (cid:12)(cid:12) k v k≤ D } ≤ min h =0 { β L − ( h ) , β L + ( h ) } . Remark . A time-periodic Lagrangian L ( x, ˙ x, t ) can be considered as an au-tonomous Lagrangian L ( θ, ˙ θ ), where ( x, ˙ x, t ) ∈ T T n × S and θ = ( t, x ). As weknow, T n × S is diffeomorphic to T n +1 . Then a time-step Lagrangian L can beconsidered as a space-step one with θ = t taken in the universal covering space R . So we just need to consider the autonomous Lagrangian of a form L ( x, ˙ x ) with( x, ˙ x ) ∈ T M in this section.We define h TL ( ¯ m , ¯ m ) = inf ¯ γ ( − T )= ¯ m ¯ γ ( T )= ¯ m Z T − T L (¯ γ ( t ) , ˙¯ γ ( t )) dt, ∀ ¯ m , ¯ m ∈ ¯ M .
From the super-linearity of L , we can see that once ¯ m and ¯ m are fixed, thereexists a finite T ¯ m , ¯ m such that h TL ( ¯ m , ¯ m ) gets its minimum. We can see thisfrom the following Lemma: Lemma 6.9. [16, 32]
If the rotation vector of each ergodic minimal measure haspositive first component π ρ ( µ ± ) > , then ∀ ¯ m = ¯ m with ¯ m ≤ < ≤ ¯ m , wehave lim T → h TL ( ¯ m , ¯ m ) = ∞ , lim T →∞ h TL ( ¯ m , ¯ m ) = ∞ . Proof.
The first formula is easy to be proved as ¯ m = ¯ m and L is super-linearof the variable ˙ x . On the other side, if there exists a sequence of T n → ∞ suchthat lim n →∞ h T n L ( ¯ m , ¯ m ) = K < ∞ , we can find ¯ γ n ∈ ¯ M as the minimizer of h T n L ( ¯ m , ¯ m ), with ¯ γ n ( − T n ) = ¯ m and ¯ γ n ( T n ) = ¯ m . Let ζ : [0 , → M be ageodesic connecting m to m , then ξ n . = ζ ∗ π ¯ γ n becomes a loop in M . ∀ ǫ > SYMPTOTIC TRAJECTORIES OF KAM TORUS 69 sufficiently small, we can always take N properly large, such that ∀ n ≥ N ,12 T n h T n L ( ¯ m , ¯ m ) = 12 T n Z T n +1 − T n L ( ξ ( t ) , ˙ ξ ( t )) dt − T n Z L + ( ζ ( t ) , ˙ ζ ( t )) dt, = 12 T n Z T n +1 − T n L + ( ξ ( t ) , ˙ ξ ( t )) dt − T n Z L + ( ζ ( t ) , ˙ ζ ( t )) dt, + 12 T n Z T n +1 − T n ( L − L + )( ξ ( t ) , ˙ ξ ( t )) dt, ≥ min h =0 β L + ( h ) − ǫ −
12 min h =0 { β L + ( h ) , β L − ( h ) } , ≥
12 min h =0 β L + ( h ) − ǫ > , as π ρ ( µ ± ) > ǫ is sufficiently small. This implies that lim n →∞ h T n L ( ¯ m , ¯ m ) = ∞ and lead to a contradiction. Then we proved the second formula and finishedthe proof. (cid:3) From the proof of this Lemma, we also get that T ¯ m , ¯ m → ∞ as − ¯ m , , ¯ m , →∞ . Since L is of a transitional form between L − and L + in the domain { ≤ x ≤ } , the following claim holds: Claim 6.10. there exists a K ′′ > such that (6.2) − K ′′ ≤ inf ≤ ¯ x , ≤ ≤ ¯ x , ≤ inf T ≥ h TL (¯ x , ¯ x ) ≤ max ≤ ¯ x , ≤ ≤ ¯ x , ≤ inf T ≥ h TL (¯ x , ¯ x ) ≤ K ′′ . Proof.
The proof of max ≤ ¯ x , ≤ ≤ ¯ x , ≤ inf T ≥ h TL (¯ x , ¯ x ) ≤ K ′′ is easy. For the other partof this claim, we can assume that there exist ¯ x and ¯ x such that inf T ≥ h TL (¯ x , ¯ x ) = −∞ . If so, there must be ¯ γ n and T n → ∞ such that h T n L (¯ x , ¯ x ) → −∞ , where¯ γ n ( − T n ) = ¯ x and ¯ γ n ( T n ) = ¯ x . But d (¯ x , ¯ x ) is bounded, we can construct asimilar loop ξ n as above Lemma and lead to a contradiction that h T n L (¯ x , ¯ x ) → + ∞ . (cid:3) Once this claim is available, we can see that(6.3) inf T ≥ h TL ( ¯ m , ¯ m ) ≥ inf ¯ x , =0 inf T − ≥ h T − L ( ¯ m , ¯ x ) + inf ¯ x , =1 inf T + ≥ h T + L (¯ x , ¯ m ) − K ′′ , and(6.4) inf T ≥ h TL ( ¯ m , ¯ m ) ≤ max ¯ x , =0 inf T − ≥ h T − L ( ¯ m , ¯ x ) + max ¯ x , =1 inf T + ≥ h T + L (¯ x , ¯ m ) + K ′′ . Then ∀{ ¯ m n } n ∈ N and { ¯ m n } n ∈ N sequences with − ¯ m n , , ¯ m n , → ∞ as n → ∞ , k inf T ≥ h TL ( ¯ m n , ¯ m n ) k and k ˙¯ γ n k is uniformly bounded for t ∈ [ − T ¯ m n , ¯ m n ], ∀ n ∈ N .That’s because we can find ¯ x n (¯ x n ) as the first (last) intersection point of ¯ γ n with { x = 0 } ( { x = 1 } ). We denote the segment of ¯ γ n from ¯ x n to ¯ x n by ¯ γ n (cid:12)(cid:12) ¯ x n ¯ x n . From(6.2), we can see that ˙¯ γ n (cid:12)(cid:12) ¯ x n ¯ x n is also uniformly bounded. On the other side, we cansee that the segment ¯ γ n (cid:12)(cid:12) ¯ x n ¯ m n (¯ γ n (cid:12)(cid:12) ¯ m n ¯ x n ) satisfies the E-L equation of L − ( L + ). Thenas n → ∞ , A L − (¯ γ n (cid:12)(cid:12) ¯ x n ¯ m n ) − h ∞ L − ( m n , y ) − h ∞ L − ( y, x n ) → † , CHONG-QING CHENG ‡ and A L + (¯ γ n (cid:12)(cid:12) ¯ m n ¯ x n ) − h ∞ L + ( x n , z ) − h ∞ L + ( z, m n ) → y ∈ M ( L − ), z ∈ M ( L + ) and x ni ∈ M ( m ni ∈ M ) is the projection of¯ x ni ( ¯ m ni ), i = 0 ,
1. Then ¯ γ n C − uniformly converges to a C − curve ¯ γ : R → ¯ M which satisfies the following: Definition 6.11.
A curve ¯ γ : R → ¯ M is contained in G ( L ) if A L (¯ γ ) (cid:12)(cid:12) [ − T,T ] = inf T ′ ∈ R + h T ′ L (¯ γ ( − T ) , ¯ γ ( T )) , ∀ T ∈ R + . We can see that G ( L ) = ∅ based on above analysis. Besides, we have: Proposition 6.12. [16]
There exists some
K > such that ∀ ¯ γ ∈ G ( L ) and T > , k h TL (¯ γ ( − T ) , ¯ γ ( T )) k ≤ K holds. Each k ∈ Z defines a Deck transformation k : ¯ M → ¯ M with k x = ( x + k, x , · · · , x n ). Let ¯ M − = { x ∈ ¯ M (cid:12)(cid:12) x ≤ } and ¯ M + = { x ∈ ¯ M (cid:12)(cid:12) x ≥ } . Definition 6.13.
A curve ¯ γ ∈ G ( L ) is called pseudo connecting curve if thefollowing holds A L (¯ γ ) (cid:12)(cid:12) [ − T,T ] = inf T ′ ∈ R + k − ¯ γ ( − T ) ∈ ¯ M − k + ¯ γ ( T ) ∈ ¯ M + h T ′ L ( k − ¯ γ ( − T ) , k + ¯ γ ( T ))for each ¯ γ ( T ) ∈ ¯ M − and ¯ γ ( T ) ∈ ¯ M + . We can denote the set of pseudo connectingcurves by C ( L ). Lemma 6.14.
The set C ( L ) is not empty.Proof. Since it’s a direct cite of [16], we only give the sketch of proof here. As weknow, G ( L ) = ∅ . So we start with a curve ¯ γ ∈ G ( L ). Given a ∆ >
0, we claimthat there are finitely many intervals [ t − i , t + i ] such that k − i ¯ γ ( t − i ) can be connectedto k + i ¯ γ ( t + i ) by another curve ζ i with ∆ − smaller action than the original one. Thisis because the previous Proposition and ∆ >
0. For a sequence ∆ i →
0, we can dofinitely many surgeries on ¯ γ and get a sequence ¯ γ i ∈ G ( L ) satisfying A L (¯ γ i ) (cid:12)(cid:12) [ − T,T ] ≤ inf T ′ ∈ R + k − ¯ γ i ( − T ) ∈ ¯ M − k + ¯ γ i ( T ) ∈ ¯ M + h T ′ L ( k − ¯ γ i ( − T ) , k + ¯ γ i ( T )) + ∆ i , ∀ T ∈ R + . On the other side, ∀ T > ∃ i such that the set { ¯ γ i (cid:12)(cid:12) [ − T,T ] : i ≥ i } is pre-compactin C ([ − T, T ] , ¯ M ). Let T → ∞ , by diagonal extraction argument ¯ γ i converges C − uniformly to a C − curve ¯ γ : R → ¯ M . Obviously, ¯ γ ∈ C ( L ). (cid:3) Theorem 6.15.
The map L → C ( L ) is upper semicontinuous. Corollary 6.16.
If the space-step Lagrangian L is periodic of x variable, then ¯ γ ∈ C ( L ) iff the projection γ = π ¯ γ : R → M is semi-static. Similar to the definition for time-step Lagrangian, we define e C ( L ) = [ ¯ γ ∈ C ( L ) (¯ γ ( t ) , ˙¯ γ ( t )) , C ( L ) = [ ¯ γ ∈ C ( L ) ¯ γ ( t ) . We can see that π e C ( L ) = e N ( L ) and π C ( L ) = N ( L ), where π : ¯ M → M is thestandard projection. SYMPTOTIC TRAJECTORIES OF KAM TORUS 71
Local connecting orbits of c-equivalence.
Recall that a time-step Lagrnagian can be considered as a space-step one fromRemark 6.8, so we only deal with the space-step case in this subsection. This newversion of c − equivalence is firstly raised in [32], which is more general than theearlier one raised in [38]. This type of connecting orbits are found in the annulus ofincomplete intersection and plays a key role in establishing transition chain crossingdouble resonance.Assume φ : T n − → T n is a smooth injection and Σ c is the image of φ . Let C ⊂ H ( T n , R ) be a connected set where we are going to define c − equivalence. Foreach class c ∈ C , we assume that there exists a non-degenerate embedded ( n − − dimensional torus Σ c ⊂ T n such that each c − semi static curve γ transversallyintersects Σ c . Let V c = \ U { i ∗ H ( U, R ) (cid:12)(cid:12) U is a neighborhood of N ( c ) ∩ Σ c } , where i : U → M is the inclusion map. V ⊥ c is defined to be the annihilator of V c ,i.e. if c ′ ∈ H ( T n , R ), h c ′ , h i = 0 for all h ∈ V c . Clearly, V ⊥ c = [ U { ker i ∗ (cid:12)(cid:12) U is a neighborhood of N ( c ) ∩ Σ c } . Note that there exists a neighborhood U of N ( c ) ∩ Σ c such that V c = i ∗ H ( U, R )and V ⊥ c = ker i ∗ [38].We say that c, c ′ ∈ H ( M, R ) are c − equivalent if there exists a continuouscurve Γ : [0 , → C such that Γ(0) = c , Γ(1) = c ′ and α (Γ( s )) keeps constant forall s ∈ [0 , ∀ s ∈ [0 , ∃ δ > s ) − Γ( s ) ∈ V ⊥ Γ( s ) whenever s ∈ [0 ,
1] and k s − s k < δ . Theorem 6.17.
Assume the cohomology class c ∗ is c − equivalent to the class c ′ through the path Γ : [0 , → H ( T n , R ) . For each s ∈ [0 , , the followings aresatisfied: • There exists at least one component of rotation vector which is positive,i.e. ∀ µ Γ( s ) ergodic Γ( s ) − minimizing measure, ω j ( µ Γ( s ) ) > for some j ∈{ , , · · · , n } . • We can find finitely many { c i } ki =1 ⊂ Γ ( c = c ∗ , c k = c ′ ) and closed 1-forms η i , ¯ µ i on M with [ η i ] = c i and [¯ µ i ] = c i +1 − c i , and smooth functions ̺ i on M j ( i ) for j = 1 , , · · · , k − such that the pseudo connecting curve se ( L i ) for the space-step Lagrangian L i = L − η i − ̺ i ¯ µ i possesses the properties: – each curve ¯ γ ∈ C ( L i ) determines an E-L orbit ( γ, ˙ γ ) of φ tL ; – the orbit ( γ, ˙ γ ) connects ˜ A ( c i ) to ˜ A ( c i +1 ) , i.e. the α − limit set is con-tained in ˜ A ( c i ) and the ω − limit set is contained in ˜ A ( c i +1 ) .Here M j ( i ) = T j ( i ) − × R × T n − j ( i ) is the covering space of M = T n , and ̺ i is a smooth map of a form ̺ i : M j ( i ) → R via ̺ i ( x ) = ̺ i ( x j ) , with ̺ i ( x j ) = 0 for x j ≤ and ̺ i ( x j ) = 1 for x j ≥ (see figure 11). † , CHONG-QING CHENG ‡ Proof.
By the definition of c − equivalence, for each c = Γ( s ) ( s ∈ [0 , ǫ > s ′ ) − c ∈ V ⊥ Γ( s ) whenever s ′ ∈ [0 ,
1] and k s − s ′ k < ǫ .Thus there exist a non-degenerately embedded ( n − − dimensional torus Σ c , aclosed form ¯ µ c and a neighborhood U of N ( c ) ∩ Σ c such that [¯ µ c ] = Γ( s ′ ) − c andsupp¯ µ c ∩ U = ∅ . We can restrict Σ c into the elementary domain T j − × [0 , × T n − j of M j and let B (Σ c , δ ) be the δ − neighborhood of Σ c in it. Then if η and ¯ µ c areclosed 1 − forms such that [ η ] = c and [ η + ¯ µ c ] = Γ( s ′ ) = c ′ , we have B (Σ c , δ ) ∩ B ( C ( L + η ) , δ ) ⊂ U, as long as δ > N ( c ) = C ( L + η ). Thenfrom the upper semicontinuity of C ( L ) w.r.t L , we have B (Σ c , δ ) ∩ B ( C ( L + η + ̺ i ¯ µ c ) , δ ) ⊂ U, as long as ̺ i ¯ µ c is sufficiently small. This is available because of the definition of c − equivalence.Besides, we can see that ¯ µ c can be chosen with its support disjoint from U . Then ∀ ¯ γ ∈ C ( L + η + ̺ i ¯ µ c ) is a solution of E-L equation determined by L , i.e. the term ̺ i ¯ µ c has no contribution to the equation along ¯ γ .From the definition of C we get that the projection of ¯ γ , which is denoted by γ ∈ M , satisfies that γ (cid:12)(cid:12) ( −∞ ,t ] is backward Γ( s ) − semi static once ¯ γ (cid:12)(cid:12) ( −∞ ,t ] fallsentirely into ¯ M − j . Similarly, we have γ (cid:12)(cid:12) [ t , + ∞ ) is forward Γ( s ′ ) − semi static once¯ γ (cid:12)(cid:12) [ t , ∞ ) falls entirely into ¯ M + j . Therefore, ( γ ( t ) , ˙ γ ( t )) → ˜ A (Γ( s )) as t → −∞ and( γ ( t ) , ˙ γ ( t )) → ˜ A (Γ( s ′ )) as t → + ∞ .Because of the compactness of [0 , s , s , · · · , s k ∈ [0 ,
1] such that above argument applies if s and s ′ are replaced respectively by s i and s i +1 . We just take c i = Γ( s i ). (cid:3) Corollary 6.18.
Let c i , η i , ¯ µ i and ̺ i be evaluated as in above Theorem, and U i be an open neighborhood of N ( c i ) ∩ Σ c i such that U i ∩ supp ¯ µ i = ∅ . Then forlarge K i > , T i > , small δ > and ∀ ¯ m, ¯ m ′ ∈ M j ( i ) with − K i ≤ ¯ m ≤− K i + 1 , K i − ≤ ¯ m ′ ≤ K i , the quantity h Tη i ,µ i ( ¯ m, ¯ m ′ ) reaches its minimum atsome T < T i and the corresponding minimizer ¯ γ i ( t, ¯ m, ¯ m ′ ) satisfies the conditionImage (¯ γ i ) ∩ B (Σ c i , δ ) .Remark . If we take L ( x, ˙ x, t ) as periodic Tonelli Lagrangian of T T n × S and M = R × T n with θ = ( t, x ) ∈ S × T n . Then the previous Theorem of c − equivalenceis just the special case discovered by Bernard in [4] and Cheng in [14, 15].With this approach, we can prove that c − equivalence of 2-resonance. Theorem 6.20. ( c − equivalence of Annulus of Incomplete Intersection) Let Γ ⊂ A ⊂ α − H ( E ) is the curve skirting around the flat F , where E ∈ (0 , ∆ ] and H is the homogenized system of 2-resonance. ∀ c, c ′ ∈ Γ is c − equivalent with eachother (see figure 25).Proof. Recall that we can consider the Lagrangian 4.47 as an autonomous one withΘ = ( X , X , S ) ∈ T . As E > ∀ c ∈ Γ, we have ω ( µ c ) = 0. Without loss ofgenerality, we can assume ω ( µ c ) >
0. Then Σ c = { X = 0 } is a 2 − dimensionalsection of T such that each c − semi static orbit intersects it transversally and N ( c ) ∩ Σ c ⊂ { ( X , X , S ) (cid:12)(cid:12) X = 0 , X ∈ ∪ I c,i } , SYMPTOTIC TRAJECTORIES OF KAM TORUS 73 where { I c,i } are finitely many intervals of T disjoint from each other. We just needto prove the equivalence for c ′ sufficiently close to c .Clearly, there exists U ⊃ N ( c ) ∩ Σ c such that V c = i ∗ H ( U, R ) = span R { (0 , , } .Then we have V ⊥ c = span R { (1 , , , (0 , , } . For each c ′ ∈ Γ sufficiently close to c , one has c ′ − c = (∆ c , ∆ c , ∈ V ⊥ c . Thus, there exists closed 1-form ¯ µ with[¯ µ ] = c ′ − c and supp ¯ µ ∩ N ( c ) ∩ Σ c = ∅ . Therefore, all classes along the curve Γ are equivalent in this case. (cid:3)
Local connecting orbits of h-equivalence.
Another type of locally connecting orbits look like heteroclinic orbits. That’sthe reason we call them type-h . This type orbits are mainly used to deal with theDiffusion problem of Arnold mechanism, which was firstly raised in [14, 15]. As thetime-periodic Lagrangian is more convenient for our application, we won’t considerit as an autonomous one in this subsection.For a Tonelli Lagrangian L ( x, ˙ x, t ) : T M × S → R with M = T in our case ofa form (4.45) or (4.46), we can assume ~e is a base vector of H ( M, N, Z ) withoutloss of generality. Then we can take ˜ M = 2 T × T n − as a finitely covering manifoldof M . Restricted to the uniform section of ˜ M × { t = 0 } , A ( L ) will become twodifferent connected components A i , i = 0 ,
1. With the approach of [17] we canconnect A to A with a semi-static heteroclinic orbits γ , of ˜ M . Besides, we canfind N i as the open neighborhood of A i , i = 0 , N , N ) > Lemma 6.21. (Connecting Lemma)
For c ∈ W g j with j = 1 , as in section 5,the Aubry set contains two different connected components A i in ˜ M and we can find N i open neighborhoods disjoint with each other containing them separately, i = 0 , .If there exists one semi-static heteroclinic orbit connecting A ( c ) to A ( c ) which isdisconnected to the others, then there exists some orbit dγ ′ of φ tL connecting ˜ A ( c ) to ˜ A ( c ′ ) for class c ′ ∈ W g j close to c . Here A i ( c ′ ) ⊂ N ∪ N , i = 0 , .Proof. This Lemma is also a direct cite of results of [14, 15, 16]. So we just give thesketch for the proof. Assume γ is the isolated heteroclinic orbit connecting A ( c )to A ( c ). We consider the modified Lagrangian L η,µ,ψ = L − η − µ − ψ, where η is a closed 1-form on T with [ η ] = c , µ is a 1-form depending on t variablein the way that µ ≡ { t ≤ } and µ = ¯ µ for { t ≥ } , where ¯ µ is a closed 1-formon T with [¯ µ ] = c ′ − c . ψ ( x, v ) : T × R → R is a smooth function with ψ ( x, t ) ≡ t ∈ ( −∞ , ∪ [1 , ∞ ). Let m, m ′ be two points in ˜ M = 2 T × T , we define h Tη,µ,ψ ( m, m ′ ) = inf γ ( − T )= mγ ( T )= m ′ Z T − T L η,µψ ( dγ ( t ) , t ) dt + α ( c ) dt, where T ∈ Z + . Note that L η,µ,ψ can be considered as a time-step Lagrangian of˜ M . Then from previous subsection of time-step Lagrangian we can define the sets C η,µ,ψ = C ( L η,µ,ψ ), C η,µ,ψ and ˜ C η,µ,ψ . Recall that C η,µ,ψ is upper semicontinuousof L and C η, , = N ( c ). Then γ ∈ N ( c ) and for c ′ sufficiently close to c and ψ † , CHONG-QING CHENG ‡ sufficiently small, there must be an orbit γ ′ ∈ C η,µ,ψ close to γ . We just need toshow that γ ′ is an orbit of φ tL by properly chosen ψ and ¯ µ .As γ is an isolated semi-static heteroclinic orbit, we can find a open and homologically-trivial ball O of it on the section { t = 0 } . Besides, no other semi-static heteroclinicorbits passing through O (see figure 26). Then we can define a non-negative func-tion f ∈ C r ( T , R ) with f ( x ) = B ( O, δ ) c , O,< ψ ( x, t ) = λρ ( t ) f ( x ) with ρ ( t ) is showed as figure 11. Here λ is just a coefficientwhich can be chosen sufficiently small. On the other side, since O is homologicallytrivial, we can choose ¯ µ with support disjoint from O . Once again we use the uppersemicontinuous of C η,µ,ψ of L we can verified γ ′ is a real orbit of φ tL . (cid:3) The orbit dγ ′ obtained in above Lemma is locally minimal in the sense we definein the following, which is crucial for the construction of globally connecting orbits. Definition 6.22. dγ : R → T M is called locally minimal orbit of type-h connecting ˜ A ( c ) to ˜ A ( c ′ ) if • dγ , is an orbit of φ tL , with the α − limit and ω − limit sets of it containedin ˜ A ( c ) and ˜ A ( c ′ ) respectively, i.e. restricted on the section { t = 0 } , α ( dγ , ) (cid:12)(cid:12) t =0 ⊂ T N and ω ( dγ , ) (cid:12)(cid:12) t =0 ⊂ T N ; • there exist two open balls V , V of ˜ M and two positive integers k − , k + such that ¯ V ⊂ N \ A ( c ), ¯ V ⊂ N \ A ( c ′ ), γ ( − k − ) ∈ V , γ ( k + ) ∈ V and h ∞ c ( x − , m ) + h k − ,k + η,µ,ψ ( m , m ) + h ∞ c ′ ( m , x + ) − lim inf k − i →∞ k + i →∞ Z k + i − k − i L η,µ,ψ ( dγ ( t ) , t ) dt − k − i α ( c ) − k + i α ( c ′ ) > m , m ) ∈ ∂ ( V × V ), x − ∈ N ∩ A ( c ) (cid:12)(cid:12) t =0 , x + ∈ N ∩A ( c ′ ) (cid:12)(cid:12) t =0 , where k ± i ∈ Z + are sequences such that γ ( − k − i ) → x − and γ ( k + i ) → x + as i → ∞ (see figure 26). Remark . Inequality (6.5) tells us that once a curve ¯ γ touches the boundary of V i , the action of L η,µ,ψ along it will be larger than the action along γ , i = 0 ,
1. As V i can be chosen arbitrarily small, it’s reasonable to call it locally minimal .6.5. Generalized transition chain.
Based on the discussion of locally connecting orbits, of c-type and h-type, nowwe can find a generalized transition chain (GTC) to verify the existence ofglobal connecting orbits, i.e. the diffusion orbits with large change of momentumvariables.Recall that the earliest definition of GTC was given by J. Mather in [38] forautonomous systems, then [4, 14, 15] generalized it to the time-periodic case. From[4] we know that if c , c ′ ∈ H ( M, R ) is equivalent with each other and there exists SYMPTOTIC TRAJECTORIES OF KAM TORUS 75
Figure 26. a GTC connecting them, then both c and c ′ lie in a flat F of α L , where L is anautonomous Lagrangian. This case is of no interest for us since ˜ A L ( c ) ∩ ˜ A L ( c ′ ) = ∅ ,this point was revealed by Massart in [35].In [32], they gave us a new way to get locally connecting orbits with a localsurgery method in a proper covering space. This skill we have illustrated in thesubsection 6.3.With these two branches of equivalent skills, globally connecting orbits then canbe constructed shadowing these locally connecting orbits. Definition 6.24.
Let c , c ′ be two classes in H ( M, R ). We say that c is joined with c ′ by a GTC if a continuous curve Γ : [0 , → H ( M, R ) exists such that Γ(0) = c ,Γ(1) = c ′ and for each s ∈ [0 ,
1] at least one of the following cases takes place: • (h-type) the Aubry set is contained in a domain N ⊂ M with nonzerotopological codimension. There exist a certain finitely covering space ˜ M ,two open domains N , N with dist( N , N ) > δ s , δ ′ s > – the Aubry set A (Γ( s )) ∩ N = ∅ , A (Γ( s )) ∩ N = ∅ and A i (Γ( s ′ )) ∩ ( N ∪ N ) = ∅ for each s, s ′ with k s − s ′ k ≤ δ s , i = 0 , – π N (Γ( s ) , ˜ M ) \B ( A i , δ ′ s ) = ∅ and there exists at least one isolated orbitin it; • (c-type) for each s ′ ∈ ( s − δ s , s + δ s ), Γ( s ′ ) is equivalent to Γ( s ). Namely,there exists a neighborhood of N (Γ( s )), which is denoted by U , such thatΓ( s ′ ) − Γ( s ) ∈ ker i ∗ . † , CHONG-QING CHENG ‡ If Γ : [0 , → H ( M, R ) is a GTC connecting c and c ′ , then we can find apartition for it with c j = { Γ( s j ) } kj =1 . Here s j ∈ [0 , c = Γ( s ), c ′ = Γ( s k ) and { , , · · · k } = { , , · · · , i } | {z } Γ i [ { i + 1 , · · · , i } | {z } Γ i − i [ · · · [ { i m − + 1 , · · · , i m } | {z } Γ im − im − with i m = k . ˜ A ( c l ) and ˜ A ( c l +1 ) can be connected by local connecting orbits ofsame type (h- or c-), as long as c i , c i +1 ∈ Γ i j − i j − , j ∈ { , , · · · , m } . This partitionalways can be found because k can be chosen sufficiently large and e N ( c ) is uppersemicontinuous of c .For our case, the GTC should be chosen in the set S ∞ i = M ( W ig ∪ A i ∪ W ig ).Since our construction is self-similar, and uniform restrictions ensure that this setis toppologically connected, we can take Γ = S ∞ i = M Γ i as a GTC and Γ i = Γ i ∪ Γ i ∪ Γ i as a partition, where Γ i ⊂ W ig , Γ i ⊂ A i and Γ i ⊂ W ig . Recall that M ≫ a posterior large enough. Γ i and Γ i is of h − type and Γ i isof c − type, this point has been showed in previous subsections. But for the validityof Connecting Lemma , we also need the existence of isolated semi-static orbitsfor all Γ i and Γ i , i = M, M + 1 , · · · , ∞ . So we need the following regularity andgenericity conditions of wNHICs of [14, 15]. Lemma 6.25. (Regularity [14, 15, 16] ) For a fixed i ∈ { M, M + 1 , · · · , ∞} , wecan take Γ i = { (0 , c ( s )) ∈ W ig (cid:12)(cid:12) c ( s ) = c ( s ′ ) for s = s ′ ∈ [0 , } and Γ i = { ( c ( s ) , ∈ W ig (cid:12)(cid:12) c ( s ) = c ( s ′ ) for s = s ′ ∈ [0 , } . Besides, we can introduce two area parameters σ j ( s ) which is one-to-one with c j ( s ) , s ∈ [0 , and j = 1 , . Then in the covering space ˜ M , we have (6.6) k u ± ,σ j ( s ) ( x ) − u ± ,σ j ( s ′ ) ( x ) k ≤ c ( q k σ j ( s ) − σ j ( s ′ ) k + k c j ( s ) − c j ( s ′ ) k ) , where i = 1 , and s, s ′ ∈ [0 , .Remark . Note that here Γ i and Γ i are not yet GTCs but only candidateones. To avoid too many symbols involved, we still denote them by these withoutambiguity.With the help of this Regularity Lemma , we can get the following genericityof isolated semi-static orbits, which is a skillful application of box-dimension.
Lemma 6.27. (Genericity [14, 15, 16] ) For a fixed i ∈ { M, M + 1 , · · · , ∞} , thesystem corresponding to W ig j with j = 1 , is of a normal form H = h + Z + R .There exists an open and dense set G i ( R ) contained in the domain B (0 , c ) ⊂ C r ( T M × S , R ) such that the system with R in it satisfies the following: For all c l ( s ) ∈ Γ i l , there exists at least one heteroclinic orbit in N ( c j ( s ) , ˜ M ) which is isolated. Here l = 1 , , s ∈ [0 , and c = c ( i ) is a proper constant depending on theunderlying resonant line. SYMPTOTIC TRAJECTORIES OF KAM TORUS 77
Based on all these preparations, we can finish our construction and get our mainconclusion now. 7.
Proof of the Main Conclusion
First, we can take proper f ( p, q, t ) ∈ C r ( T ∗ T × S , R ) for system (1.2), thentransform it into (1.4) with an exact symplectic transformation R ∞ f in a smallneighborhood D of { } × T × S . Here f ( q, p, t ) can be chosen satisfying all the U* and C* conditions and denoted by f = f ( q, p, t ). Then we get a connectedcohomology set S ∞ i = M ( W ig ,f ∪ A if ∪ W ig ,f ), of which the wNHIC N ig j ,f cor-responding to W ig i ,f persists with certain length and the thickness of A if can beuniformly estimated, j = 1 ,
2. This point is based on our analysis in section 4 and 5.Second, we choose the candidate GTCs satisfying Γ i ,f ⊂ W ig ,f , Γ i ,f ⊂ A if and Γ i ,f ⊂ W ig ,f . Then we can add a small perturbation ǫ ∆ f ( p, q, t ) to f and f = f + ǫ ∆ f . Here ∆ f ( q, p, t ) ∈ B r + r ′ (0 , ⊂ C r + r ′ ( T ∗ T × S , R ), ǫ (0 < ǫ ≪
1) is a small constant and r ′ > Z ∆ f of H . = H + ǫ ∆ f along the resonant plan P f will have much smaller Fourier coefficients, as longas i ≥ M ≫ r ′ is properly large. This point is very important for ourconstruction: H still satisfies the uniform restrictions U* and C* . Additionally,Γ i ,f is now a GTC of H from Lemma 6.27. Besides, the thickness of A if canpersist with just a ǫ ∆ decrease from Corollary 5.8. So Γ i ,f ⊂ A if is also a GTC.After above once perturbation with ǫ ∆ f , we can then add another pertur-bation ǫ ∆ f to H and get H . = H + ǫ ∆ f . Here 0 < ǫ ≪ ǫ and ∆ f ∈B r + r ′ (0 , H andΓ i ,f ∪ Γ i ,f ∪ Γ i ,f is now a GTC contained in W ig ,f ∪ A if ∪ W ig ,f . So we havefound a GTC for a whole transport process Γ ωi .Repeat above process and we get H ∞ = H + P ∞ k =1 ǫ k ∆ f k . For this system H ∞ , S ∞ i = M (Γ i ,f ∞ ∪ Γ i ,f ∞ ∪ Γ i ,f ∞ ) then become a whole GTC and we finally find theasymptotic trajectories for T ω torus.8. Appendix
Introduction of Melnikov Method.
This part serves as a supplement of Proposition 5.5, which can be found ofanother version in [50]. But for the completeness of this paper, we list it in thefollowing.For a uncoupled system H ( x, q, y, p ) = H , ( x, y ) + H , ( q, p ) with ( x, q, y, p ) ∈ T ∗ T , we assume (0 , , ,
0) is the unique hyperbolic fixed point. Without lossof generality, we let H (0 , , ,
0) = 0. Let H ǫ ( x, q, y, p ) = H + ǫH ( x, q, y, p ) † , CHONG-QING CHENG ‡ be a perturbed system where ǫ ≪
1. Then we can still get a unique hyper-bolic fixed point for H ǫ , which can be denoted by ( x ∗ ( ǫ ) , q ∗ ( ǫ ) , y ∗ ( ǫ ) , p ∗ ( ǫ )). Wecan see that ( x ∗ (0) , q ∗ (0) , y ∗ (0) , p ∗ (0)) = (0 , , , H ǫ ( x ∗ ( ǫ ) , q ∗ ( ǫ ) , y ∗ ( ǫ ) , p ∗ ( ǫ )) = 0.As H is a uncoupled system with two first integrals H , and H , , we canfind the generating functions S u,s ( x, q ) defined in a proper neighborhood Ω of(0 , ∈ T . Besides, we can assume that suspended in the universal coveringspace, the union of all the copies of Ω can cover R . The trajectory with initialposition ( x, q, ∂S u ( x,q ) ∂x , S u ( x,q ) ∂q ) (or ( x, q, ∂S s ( x,q ) ∂x , S s ( x,q ) ∂q )) will tend to (0 , , ,
0) as t → −∞ (or t → ∞ ). Here ( q, x ) ∈ Ω is a fixed point. We can also see thatthe whole trajectory will lay on the graph { ( q, x, ∂S u ( x,q ) ∂x , S u ( x,q ) ∂q ) (cid:12)(cid:12) ( q, x ) ∈ Ω } (or { ( x, q, ∂S s ( x,q ) ∂x , S s ( x,q ) ∂q ) (cid:12)(cid:12) ( q, x ) ∈ Ω } ).As ǫ ≪
1, we can still get the persistent generating functions S u,sǫ for H ǫ in thedomain Ω. Formally we take S u,sǫ = S u,s + ǫS u,s + O ( ǫ ), then we have0 = H ǫ ( x, q, ∇ S u,sǫ ( x, q ))= H ( x, q, ∇ S u,s + ǫ ∇ S u,s ) + ǫH ( x, q, ∇ S u,s ) + O ( ǫ )= ǫ ( ∂H ∂y , ∂H ∂p ) (cid:12)(cid:12) ( x,q, ∇ S u,s ) · ∂S u,s ∂x∂S u,s ∂q ! + ǫH ( x, q, ∇ S u,s ) + O ( ǫ ) . We denote by ( x u,s ( t ) , q u,s ( t ) , y u,s ( t ) , p u,s ( t )) the trajectory of H with initial po-sition ( x u,s (0) , q u,s (0) , y u,s (0) , p u,s (0)) = ( x, q, ∂S u,s ∂x , ∂S u,s ∂q ). As H is uncoupled,these two trajectories can actually be joint into a whole γ ( t ) = ( x u,s ( t ) , q u,s ( t ))with t ∈ R . We can omit the superscript ‘u,s’ for short.The O ( ǫ ) term of above formula is of a form: ddt S u,s ( γ ( t )) + H ( γ ( t ) , ∇ S u,s ( γ ( t ))) = 0 . Then we take a path integral by S u ( x ( t ) , q ( t )) (cid:12)(cid:12) −∞ = − R −∞ H ( γ ( t ) , ∇ S u,s ( γ ( t ))) dt,S s ( x ( t ) , q ( t )) (cid:12)(cid:12) ∞ = − R ∞ H ( γ ( t ) , ∇ S u,s ( γ ( t ))) dt, where ( x (0) , q (0)) = ( x, q ). On the other side, we have S u,s (0 ,
0) = 0. This isbecause we can make S u,sǫ ( x ∗ ( ǫ ) , q ∗ ( ǫ )) ≡ S u,s ( x ∗ ( ǫ ) , q ∗ ( ǫ )) ∼ O ( ǫ ) , ∀ ǫ ≪ , as ( x ∗ ( ǫ ) , q ∗ ( ǫ )) = (0 ,
0) + ǫ ( x ∗ , x ∗ ) + O ( ǫ ) is formally valid. So we get the O ( ǫ )term S u,s (0 ,
0) = 0.With the help of above relationships, we get the Melnikov function by M ( x, q ) . = S u ( x, q ) = S s ( x, q ) = − Z ∞−∞ H ( γ ( t ) , ∇ S u,s ( γ ( t ))) dt. SYMPTOTIC TRAJECTORIES OF KAM TORUS 79
We can easily see that this function is invariant with respect to the flow of H , i.e. ∇ H M ≡
0. Since H = H , + H , , we then get ∇ H , M ≡ −∇ H , M , ∀ ( x, q ) ∈ T .For a fixed ( x, q ) ∈ T satisfying ∇ H , M ( x, q ) = ∇ H , M ( x, q ) = 0, we can geta 2 × ∇ H ,i ∇ H ,j M ) i,j =1 , , whose rank is at most 1. If so, there mustbe a unique homoclinic point ( x ( ǫ ) , q ( ǫ )) in a O ( ǫ ) − neighborhhod of ( x, q ).8.2. A generalization of U2 condition.
In this subsection, we can generalize our condition U2 to a loose one. Lemma 8.1. [16]
Let { F λ ( x ) : T → R } be a family of C r − functions ( r ≥ ) with λ contained in [ λ , λ ] . If F λ is Lipschitz continuous w.r.t. λ , then we can find anopen dense set G ⊂ C r ( T , R ) such that ∀ V ∈ G , the followings hold: • (ND) ∀ λ ∈ [ λ , λ ] , all the maximizers of F λ ( x ) are non-degenerate. • ∀ λ ∈ [ λ , λ ] , there exist at most two maximizers for F λ ( x ) . • there exist finitely many λ i ∈ ( λ , λ ) such that F λ i ( x ) has two differentmaximizers.Remark . We call such a λ i a bifurcation point . If we modify the first bulletby the following: (UND) ∀ λ ∈ [ λ , λ ], all the maximizers of F λ ( x ) is uniformly non-degeneratewith the eigenvalues not bigger than − c ∗ < p σ [ f ] takes place of F λ with p ∈ S as a parameter. We can justtake c ∗ = c d σm l m ( r +2) . Here ‘m’ reminds us of which resonant line we are considering.As we have showed that the set of functions satisfying U2 is non-empty, so theopen dense property of above Lemma ensures that the functions satisfying thesethree bullets and (UND) do exist. Acknowledgement
This work is supported by NNSF of China (Granted 11171146,Granted 11201222), National Basic Research Program of China (973 Program,2013CB834100) and Basic Research Program of Jiangsu Province (BK2008013).The first author is grateful to C-Q Cheng for teaching his work in a priori stableArnold Diffusion. Besides, he also thanks Ji Li for checking the details of this paper,and thanks Jinxin Xue for proposing the result of Wiggins to me, which does helpme a lot in solving the 2-resonance difficulties.Both the authors thank J. Cheng, W. Cheng, J. Yan, X. Cui, M. Zhou and allthe other colleagues in Dynamical Systems seminar of Nanjing University. Theyreally gave us several inspiring discussions in the process of this paper. † , CHONG-QING CHENG ‡ References [1] Arnold V.
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